Stable homeomorphism conjecture for torsion-free fundamental groups

Let $M$ and $N$ be closed, orientable, topological $4$-manifolds that are homotopy equivalent, have torsion-free fundamental group, and have the same Kirby–Siebenmann invariant. Stable homeomorphism conjecture. Then $M$ and $N$ are stably homeomorphic. The conjecture follows in the simply connected case from Freedman's work and holds when the universal cover is not Spin; the general torsion-free case remains open.