Contactomorphism status of boundary diffeomorphisms for Stein corks

Determine whether the boundary diffeomorphisms of the Stein corks C(1,1;-1) and C_m described in Gompf's construction and in Section 2 are contactomorphisms, thereby establishing whether these Stein corks are contact corks.

Background

The paper constructs Gompf's infinite order cork C(1,1;-1) and an infinite family of modified infinite order corks C_m, proving that all of them admit Stein structures. A contact cork additionally requires the relevant boundary diffeomorphism to preserve the contact structure induced on the boundary of the Stein domain. The paper explicitly states that it remains unresolved whether the boundary diffeomorphisms associated with these corks have this property.