The solid-torus exhaustion problem for positive scalar curvature

Let $M^3$ be a three-manifold that has an exhaustion by solid tori and admits a complete metric with scalar curvature $R>0$ and bounded geometry. The solid-torus exhaustion problem. Must $M^3$ be the interior of a genus-one handlebody? This asks whether the stated exhaustion and geometric conditions force the standard solid-torus-interior topology. It is presented as a problem suggested by the authors' work, and the source gives no resolution evidence.