Concordance-implies-isotopy conjecture for positive scalar curvature metrics

Let $M$ be a closed manifold, and let $\mathrm{Riem}^+(M)$ denote the space of positive scalar curvature metrics on $M$. Two metrics in this space are concordant if there is a positive scalar curvature metric on $M\times[-1,1]$ restricting near the endpoints to their product metrics, and isotopic if they are joined by a smooth path. Concordance-implies-isotopy conjecture. If $g_0,g_1\in\mathrm{Riem}^+(M)$ are concordant, then they are also isotopic. Concordance always implies isotopy only in the conjectural converse direction; whether every concordance can be replaced by an isotopy remains open.