A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds

Let $(X^{n+1},g_+)$ be a Poincar\'e--Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe type. We prove the sharp relative comparison inequality $$\frac{Y_1(X,M,[\bar g])}{Y_1(\mathbb{S}^{n+1}_+,\mathbb{S}^n,[g_{\mathbb{S}_+^{n+1}}])} \geq \left(\frac{Y(M,[h])}{Y(\mathbb{S}^n,[g_{\mathbb{S}^n}])}\right)^{\frac{n}{n+1}}$$ for the type-I Escobar--Yamabe compactification, and establish the rigidity. This confirms a conjecture proposed by Sun-Yung A. Chang.