Chen's smooth product conjecture for elliptic 3-manifold s-cobordisms

Let $M$ be an elliptic $3$-manifold, and let $W$ be a smooth $s$-cobordism from $M$ to itself. Let $\widetilde{W}$ be its universal cover. Chen's smooth product conjecture. $W$ is smoothly a product if and only if $\widetilde{W}$ is smoothly a product. This conjecture is the central proposal of the paper's program for understanding smooth $s$-cobordisms of elliptic $3$-manifolds; it suggests that any nontrivial smooth $s$-cobordism, if one exists, is unrelated to the fundamental group of the $3$-manifold. The paper's main theorem establishes the analogous product result for symplectic $s$-cobordisms under a canonical contact-structure assumption, while the smooth conjecture remains open.