diff --git a/typed-racket-lib/typed-racket/private/type-contract.rkt b/typed-racket-lib/typed-racket/private/type-contract.rkt
index 4e9cc8c9..58242cd6 100644
--- a/typed-racket-lib/typed-racket/private/type-contract.rkt
+++ b/typed-racket-lib/typed-racket/private/type-contract.rkt
@@ -17,7 +17,7 @@
  racket/format
  syntax/flatten-begin
  (only-in (types abbrev) -Bottom -Boolean)
- (static-contracts instantiate optimize structures combinators constraints)
+ (static-contracts instantiate structures combinators constraints)
  (only-in (submod typed-racket/static-contracts/instantiate internals) compute-constraints)
  ;; TODO make this from contract-req
  (prefix-in c: racket/contract)
@@ -287,15 +287,14 @@
                         #:sc-cache [sc-cache (make-hash)])
   (let/ec escape
     (define (fail #:reason [reason #f]) (escape (init-fail #:reason reason)))
-    (instantiate
-     (optimize
-      (type->static-contract ty #:typed-side typed-side fail
-                             #:cache sc-cache)
-      #:trusted-positive typed-side
-      #:trusted-negative (not typed-side))
+    (instantiate/optimize
+     (type->static-contract ty #:typed-side typed-side fail
+                            #:cache sc-cache)
      fail
      kind
-     #:cache cache)))
+     #:cache cache
+     #:trusted-positive typed-side
+     #:trusted-negative (not typed-side))))
 
 (define any-wrap/sc (chaperone/sc #'any-wrap/c))
 
diff --git a/typed-racket-lib/typed-racket/static-contracts/instantiate.rkt b/typed-racket-lib/typed-racket/static-contracts/instantiate.rkt
index e437e1cf..c5c1173c 100644
--- a/typed-racket-lib/typed-racket/static-contracts/instantiate.rkt
+++ b/typed-racket-lib/typed-racket/static-contracts/instantiate.rkt
@@ -15,6 +15,7 @@
   "combinators/case-lambda.rkt"
   "combinators/parametric.rkt"
   "kinds.rkt"
+  "optimize.rkt"
   "parametric-check.rkt"
   "structures.rkt"
   "constraints.rkt"
@@ -23,33 +24,53 @@
 (provide static-contract-may-contain-free-ids?)
 
 (provide/cond-contract
+ [instantiate/optimize
+     (parametric->/c (a) ((static-contract? (-> #:reason (or/c #f string?) a))
+                          (contract-kind? #:cache hash? #:trusted-positive boolean? #:trusted-negative boolean?)
+                          . ->* . (or/c a (list/c (listof syntax?) syntax?))))]
  [instantiate
      (parametric->/c (a) ((static-contract? (-> #:reason (or/c #f string?) a))
-                          (contract-kind? #:cache hash?)
+                          (contract-kind? #:cache hash? #:recursive-kinds (or/c hash? #f))
                           . ->* . (or/c a (list/c (listof syntax?) syntax?))))]
  [should-inline-contract? (-> syntax? boolean?)])
 
-
 ;; Providing these so that tests can work directly with them.
 (module* internals #f
   (provide compute-constraints
            compute-recursive-kinds
            instantiate/inner))
 
+(define (instantiate/optimize sc fail [kind 'impersonator] #:cache [cache #f] #:trusted-positive [trusted-positive #f] #:trusted-negative [trusted-negative #f])
+  (define recursive-kinds
+    (with-handlers [(exn:fail:constraint-failure?
+                    (lambda (exn)
+                      ;; Even if the constraints for `sc` are unsolvable,
+                      ;;  the optimizer might be able to reduce parts of
+                      ;;  `sc` to give a contract with solvable constraints.
+                      ;; This currently happens for the `Any-Syntax` type;
+                      ;;  eventually that won't happen for `Any-Syntax`,
+                      ;;  and at that point maybe we can fail here. -- Ben G.
+                      #f))]
+      (compute-recursive-kinds
+        (contract-restrict-recursive-values (compute-constraints sc kind)))))
+  (define sc/opt (optimize sc #:trusted-positive trusted-positive #:trusted-negative trusted-negative #:recursive-kinds recursive-kinds))
+  (instantiate sc/opt fail kind #:cache cache #:recursive-kinds recursive-kinds))
+
 ;; kind is the greatest kind of contract that is supported, if a greater kind would be produced the
 ;; fail procedure is called.
 ;;
 ;; The cache is used to share contract definitions across multiple calls to
 ;; type->contract in a given contract fixup pass. If it's #f then that means don't
 ;; do any sharing (useful for testing).
-(define (instantiate sc fail [kind 'impersonator] #:cache [cache #f])
+(define (instantiate sc fail [kind 'impersonator] #:cache [cache #f] #:recursive-kinds [recursive-kinds #f])
   (if (parametric-check sc)
       (fail #:reason "multiple parametric contracts are not supported")
       (with-handlers [(exn:fail:constraint-failure?
                         (lambda (exn) (fail #:reason (exn:fail:constraint-failure-reason exn))))]
         (instantiate/inner sc
-          (compute-recursive-kinds
-            (contract-restrict-recursive-values (compute-constraints sc kind)))
+          (or recursive-kinds
+              (compute-recursive-kinds
+                (contract-restrict-recursive-values (compute-constraints sc kind))))
           cache))))
 
 ;; computes the definitions that are in / used by `sc`
diff --git a/typed-racket-lib/typed-racket/static-contracts/optimize.rkt b/typed-racket-lib/typed-racket/static-contracts/optimize.rkt
index aec98295..56638c9f 100644
--- a/typed-racket-lib/typed-racket/static-contracts/optimize.rkt
+++ b/typed-racket-lib/typed-racket/static-contracts/optimize.rkt
@@ -17,7 +17,7 @@
 
 
 (provide/cond-contract
- [optimize ((static-contract?) (#:trusted-positive boolean? #:trusted-negative boolean?)
+ [optimize ((static-contract?) (#:trusted-positive boolean? #:trusted-negative boolean? #:recursive-kinds (or/c #f hash?))
                                . ->* . static-contract?)])
 
 ;; Reduce a static contract to a smaller simpler one that protects in the same way
@@ -110,15 +110,15 @@
 
 
 ;; Reduce a static contract assuming that we trusted the current side
-(define (trusted-side-reduce sc)
+(define ((make-trusted-side-reduce flat-sc?) sc)
   (match sc
     [(->/sc: mand-args opt-args mand-kw-args opt-kw-args rest-arg (list (any/sc:) ...))
      (function/sc #t mand-args opt-args mand-kw-args opt-kw-args rest-arg #f)]
     [(arr/sc: args rest (list (any/sc:) ...))
      (arr/sc args rest #f)]
     [(none/sc:) any/sc]
-    [(or/sc: (? flat-terminal-kind?) ...) any/sc]
-    [(? flat-terminal-kind?) any/sc]
+    [(or/sc: (? flat-sc?) ...) any/sc]
+    [(? flat-sc?) any/sc]
     [(syntax/sc: (? recursive-sc?))
      ;;bg; _temporary_ case to allow contracts from the `Syntax` type.
      ;;    This is temporary until TR has types for immutable-vector
@@ -126,9 +126,6 @@
      any/sc]
     [else sc]))
 
-(define (flat-terminal-kind? sc)
-  (eq? 'flat (sc-terminal-kind sc)))
-
 ;; The side of a static contract describes the source of the values that
 ;;  the contract needs to check.
 ;; - 'positive : values exported by the server module
@@ -176,12 +173,13 @@
 ;; update-side : sc? weak-side? -> weak-side?
 ;; Change the current side to something safe & strong-as-possible
 ;;  for optimizing the sub-contracts of the given `sc`.
-(define (update-side sc side)
+(define ((make-update-side flat-sc?) sc side)
   (match sc
    [(or/sc: scs ...)
-    #:when (not (andmap flat-terminal-kind? scs))
+    #:when (not (andmap flat-sc? scs))
     (weaken-side side)]
-   [(? guarded-sc?)
+   [_
+    #:when (guarded-sc? sc)
     (strengthen-side side)]
    [_
     ;; Keep same side by default.
@@ -194,8 +192,7 @@
 ;;  type constructor. E.g. list/sc is "real" and or/sc is not.
 (define (guarded-sc? sc)
   (match sc
-   [(or (? flat-terminal-kind?)
-        (->/sc: _ _ _ _ _ _)
+   [(or (->/sc: _ _ _ _ _ _)
         (arr/sc: _ _ _)
         (async-channel/sc: _)
         (box/sc: _)
@@ -292,9 +289,28 @@
         (sc-map sc trim)]))
   (trim sc 'covariant))
 
+(define (make-sc->kind recursive-kinds)
+  (if recursive-kinds
+    (λ (sc)
+      (let loop ([sc sc])
+        (match sc
+         [(recursive-sc _ _ body)
+          (loop body)]
+         [(or (recursive-sc-use id)
+              (name/sc: id))
+          (hash-ref recursive-kinds id #f)]
+         [_
+          (sc-terminal-kind sc)])))
+    sc-terminal-kind))
 
 ;; If we trust a specific side then we drop all contracts protecting that side.
-(define (optimize sc #:trusted-positive [trusted-positive #f] #:trusted-negative [trusted-negative #f])
+(define (optimize sc #:trusted-positive [trusted-positive #f] #:trusted-negative [trusted-negative #f] #:recursive-kinds [recursive-kinds #f])
+  (define flat-sc?
+    (let ([sc->kind (make-sc->kind recursive-kinds)])
+      (λ (sc) (eq? 'flat (sc->kind sc)))))
+  (define trusted-side-reduce (make-trusted-side-reduce flat-sc?))
+  (define update-side (make-update-side flat-sc?))
+
   ;; single-step: reduce and trusted-side-reduce if appropriate
   (define (single-step sc maybe-weak-side)
     (define trusted
diff --git a/typed-racket-test/succeed/issue-628.rkt b/typed-racket-test/succeed/issue-628.rkt
new file mode 100644
index 00000000..c6c6a1f3
--- /dev/null
+++ b/typed-racket-test/succeed/issue-628.rkt
@@ -0,0 +1,22 @@
+#lang racket/base
+
+;; Test that the `Spec` type can be converted to a contract that
+;;  successfully **runs**
+
+(module t typed/racket/base
+  (provide f g)
+
+  (define-type Spec
+    (-> (U Spec String)))
+
+  (: f (-> Spec))
+  (define (f)
+    (λ () "hello"))
+
+  (: g (-> (Rec T (-> (U T String)))))
+  (define (g)
+    (λ () "hello")))
+
+(require 't)
+
+(void f g)