The Green's function mass conjecture for the fractional Yamabe problem

Let $(X,g_+)$ be a conformally compact Einstein manifold with compactification $(\overline X,\overline g)$ and boundary $N$, and suppose that $\Lambda^\gamma(N,[\overline g|_N])>0$. Let $G_p$ be the positive Green's function associated to a point $p\in N$. If either $(N,[\overline g|_N])$ is locally conformally flat or $n=2$, then, for $q\in\overline X$ near $p$, $G_p$ has the expansion

$$G_p(q)=c_{n,\gamma}d_{\overline g}(q,p)^{2\gamma-n}+A+O'(d_{\overline g}(q,p)^{\min\{1,2\gamma\}}),$$

where $A\in\mathbb R$, $c_{n,\gamma}>0$, and $d_{\overline g}(\cdot,\cdot)$ is the distance function. Green's function mass conjecture. The constant $A$ satisfies $A\geq 0$, with equality if and only if $(\overline X,\overline g)$ is conformally diffeomorphic to the standard unit ball in $\mathbb R^{n+1}$. The assertion is presented in the context of the fractional Yamabe problem and is attributed to Kim, Musso, and Wei; no resolution is supplied in the source.