Schoen's conjecture on singular metrics with nonnegative scalar curvature

Let $M$ be a closed smooth manifold, and let $g$ be an $L^\infty$-metric on $M$, meaning that for some smooth reference Riemannian metric $g_0$ and constant $\Lambda>0$, $\Lambda^{-1}g_0\leq g\leq\Lambda g_0$ almost everywhere. Let $S\subset M$ be a closed embedded submanifold of codimension at least $3$, and suppose that $g$ is smooth on $M\setminus S$. Assume that $\operatorname{Sc}(g)\geq 0$ on $M\setminus S$ and that $M$ has nonnegative Yamabe invariant. Schoen's conjecture. The metric $g$ extends to a smooth flat metric on $M$. As stated in this level of generality, the conjecture is false, as shown by counterexamples of Cecchini--Frenck--Zeidler.