The Doubling Conjecture for positive scalar curvature

Let $X$ be a compact manifold with boundary, let $M=\operatorname{Dbl}(X,\partial X)$ be its double, and let a metric of positive scalar curvature mean a Riemannian metric whose scalar curvature is everywhere positive. A metric on $X$ has positive mean curvature on $\partial X$ when the mean curvature of the boundary, with respect to the chosen outward normal, is positive. Doubling Conjecture. If $M$ admits a metric of positive scalar curvature, then $X$ admits a metric of positive scalar curvature with positive mean curvature on $\partial X$. This is the proposed converse to the paper's doubling theorem and asks when positive scalar curvature on the double can be detected by a suitable metric on the manifold with boundary; its status is presented as an open problem.