The handlebody positive-scalar-curvature conjecture

Let $M^3$ be the interior of a genus-$\gamma$ handlebody, equipped with a complete metric whose scalar curvature satisfies $R>0$. The handlebody positive-scalar-curvature conjecture. Then $\gamma\leq 1$. This seeks a topological classification of three-manifold interiors admitting complete metrics of positive scalar curvature; the genus-one case is known to occur. The source says the conjecture is proved under bounded-geometry assumptions, but gives no resolution in general.