Nonexistence of Stein structures on the contractible 4-manifolds $X_{n,k}$

Let $X_{n,k}$ be the compact contractible oriented smooth $4$-manifold whose boundary is Stein fillable. Nonexistence conjecture. The manifold $X_{n,k}$ does not admit any Stein structure for $n\geq 2$ and $k\geq 4n$. This proposes potential counterexamples to the problem of whether every compact contractible smooth $4$-manifold with Stein fillable boundary admits a Stein structure; the status of the conjecture is not resolved in the supplied text.