The universal-cover criterion for smooth triviality of elliptic 3-manifold s-cobordisms

An elliptic 3-manifold is a quotient of $\mathbb S^3$ by a free action of a finite subgroup of $SO(4)$. A smooth $s$-cobordism of elliptic $3$-manifolds is a smooth $4$-manifold whose boundary is the disjoint union of two elliptic $3$-manifolds and for which the inclusion of each boundary component is a homotopy equivalence. It is smoothly trivial when it is diffeomorphic to the product of an elliptic $3$-manifold with $[0,1]$.

Universal-cover criterion. A smooth $s$-cobordism of elliptic $3$-manifolds is smoothly trivial if and only if its universal cover is smoothly trivial.

This conjecture addresses whether trivial topological $s$-cobordisms of elliptic $3$-manifolds can carry exotic smooth structures and whether nontrivial topological examples are smoothable. Its resolution is part of a proposed geometrization program for these smooth $s$-cobordisms.