Let $M_0^4$ and $M_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M_0$ and $M_1$ and with the inclusions of $M_0$ and $M_1$ into $M$ both being homotopy equivalences). In the case where $M_0$ and $M_1$ are simply-connected there is a result due to Curtis, Hsiang, Freedman, and Stong that says that actually there is a manifold $A \subset M$ that is an h-cobordism between $A_0 := A \cap M_0$ and $A_1 := A \cap M_1$ (here $A$ is a manifold with corners) such that $A_0$ and $A_1$ are contractible manifolds and $M - int(A)$ is a product cobordism. Further conditions where later added on (for example, that $A$ may be chosen so that $M-A$ is simply-connected) - see here for further discussion.

I am wondering if a similar result exists in the case of $h$-cobordisms between non-simply-connected 4-manifolds. Does anyone know of such a result or have ideas on why such a result should not exist?