The trivial h-cobordism conjecture for torsion-free fundamental groups

Let $M$ be a closed, connected, smooth manifold of dimension at least $5$ with torsion-free fundamental group. An $h$-cobordism over $M$ is a smooth cobordism over $M$ for which both boundary inclusions are homotopy equivalences; it is trivial if it is isomorphic to the cylinder $M\times[0,1]$. The trivial h-cobordism conjecture. Every $h$-cobordism over $M$ is trivial. By the $s$-Cobordism Theorem, this is equivalent for a fixed $M$ to vanishing of the Whitehead group of $\pi_1(M)$; the source gives no resolution in general.