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

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