Fractional weighted isoperimetric ratio strict inequality conjecture

Let $n$ be the boundary dimension, let $\gamma\in(0,\frac{1}{2}]$ with $n>2\gamma$, and let $(\overline{X},\bar{g})$ be a smooth compactification that is not conformally diffeomorphic to the Euclidean unit ball $(\mathbb{B}^N,g_{\mathbb{R}^N})$. Let $\Theta_{n,\gamma}(X,g^+,[\bar{h}])$ be the corresponding fractional weighted isoperimetric variational problem, and let $\Theta_{n,\gamma}$ and $\theta_{n,\gamma}$ denote the quantities appearing in the source's inequality. Fractional weighted isoperimetric conjecture. If $n>2\gamma$ and $(\overline{X},\bar{g})$ is not conformally diffeomorphic to $(\mathbb{B}^N,g_{\mathbb{R}^N})$, then holds. Consequently, the variational problem $\Theta_{n,\gamma}(X,g^+,[\bar{h}])$ is attainable. This extends the preceding strict-inequality expectation to the fractional weighted setting; the source presents it as a proposed conjecture, and no resolution is supplied.