Gompf's standardness conjecture for twisted doubles of infinite order corks

Let $C=C(r,s;m)$ be Gompf's cork, and define the twisted double

$$\mathbb S_{r,s,m,k}:=C\cup_{f^k}(-C)=S^4(C,f^k).$$

Here $r,s,m$ are integers with $r,s>0>m$, and $k$ is a nonzero integer. Gompf's conjecture. The homotopy 4-sphere $\mathbb S_{r,s,m,k}$ is diffeomorphic to the standard $S^4$. The untwisted double is known to be diffeomorphic to the standard $S^4$, while the differential structures of the twisted doubles are difficult to distinguish; the statement concerns all nonzero twists.