Chang's sharp relative comparison conjecture for Poincaré–Einstein manifolds

Let $(X^N,g_+)$, with $N=n+1\ge 3$, be a smooth Poincaré–Einstein manifold with conformal infinity $(M^n,[h])$. Assume that $Y(M,[h])>0$, and let $(\overline X,[\bar g])$ denote its conformal compactification. Then Chang's sharp relative comparison conjecture.

$$\frac{Y_1(\overline X,M,[\bar g])}{Y_1(\mathbb S^{n+1}_+,\mathbb S^n,[g_{\mathbb S^{n+1}_+}])} \ge \left(\frac{Y(M,[h])}{Y(\mathbb S^n,[g_{\mathbb S^n}])}\right)^{\frac{n}{n+1}}.$$

Equality holds if and only if $(X^N,g_+)$ is isometric to $(\mathbb H^N,g_{\mathbb H^N})$. The exponent $\frac{n}{n+1}$ is forced by scaling. The conjecture proposes the sharp conformally invariant comparison inequality motivated by earlier inequalities of Gursky–Han and the type-II Escobar–Yamabe compactification; the general validity and rigidity statement remain to be established.