The contractible three-manifold positive-scalar-curvature conjecture

Let $(M^3,g)$ be a contractible $3$-manifold equipped with a complete metric whose scalar curvature satisfies $R>0$. The contractible three-manifold positive-scalar-curvature conjecture. Then $M$ is diffeomorphic to $\mathbf{R}^3$. The question concerns whether positive scalar curvature can occur on exotic contractible open three-manifolds. The source states that the conjecture is proved under bounded-geometry assumptions, but gives no resolution in general.