Schoen's Weyl-vanishing conjecture for Yamabe blow-up points

Let $(M,g_0)$ be a Riemannian manifold of dimension $n\geq 6$, and let $g_\nu=u_\nu^\frac{4}{n-2}g_0$ be a sequence of solutions to the Yamabe problem. A point $\overline{y}\in M$ is a blow-up point if it is a blow-up point of this sequence. Set

$$d=\left[\frac{n-2}{2}\right].$$

Schoen's Weyl-vanishing conjecture. If $\overline{y}$ is a blow-up point, then the Weyl tensor $W_{g_0}$ of $g_0$ vanishes at $\overline{y}$ to order $\left[\frac{n-6}{2}\right]$; equivalently,

$$\limsup_{y\to\overline{y}}d_{g_0}(\overline{y},y)^{2-d}\lvert W_{g_0}(y)\rvert=0.$$

This conjecture concerns the geometric restrictions imposed on possible blow-up points in the Yamabe problem. The compactness conjecture was known in the locally conformally flat and three-dimensional cases, and had been proved in dimensions $4$ and $5$; dimensions $n\geq6$ were described as more subtle because the associated asymptotically flat metric need not have a well-defined ADM mass without suitable Weyl-tensor vanishing.