Existence of an infinite-order cork satisfying the Stein condition

Determine whether there exists a pair (W, f) where W is a smooth contractible Stein 4-manifold and f: ∂W → ∂W is a boundary diffeomorphism of infinite order such that cutting W out of a smooth 4-manifold and regluing by iterates f^n changes the smooth structure for infinitely many n; equivalently, establish the existence of an infinite-order cork with the Stein condition.

Background

The note distinguishes between a loose cork (a contractible submanifold W with a boundary involution f that changes the smooth structure upon regluing) and a cork, which additionally requires W to be Stein.

It is known that by weakening the involution condition on the boundary map, one can construct infinite-order loose corks; examples are given in Akbulut (2017) and Gompf (2016). However, the existence of an infinite-order analogue that also satisfies the Stein condition remains unresolved.