In a sense, this is a follow-up to this question.

By work of Freedman and Wall, it is known that if two *simply-connected* 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ such that $M \times \#^k (S^2 \times S^2)$ is *diffeomorphic* to $N \times \#^k (S^2 \times S^2)$.
(By $\#^k$, we denote $k$-fold connected sum.)

Can this statement be generalised to *non-simply-connected* manifolds, possibly including Whitehead torsion?