The strict isoperimetric-ratio conjecture for conformal manifolds with boundary

Let $n\geq 3$, let $(M^n,g)$ be a smooth compact Riemannian manifold with nonempty boundary, and suppose that $\lambda_1(L_g)>0$. For a conformal metric $\widetilde g\in[g]$ with zero scalar curvature, let $I(M,\widetilde g)$ denote the isoperimetric ratio and define

$$\Theta_{M,g}=\sup\{I(M,\widetilde g):\widetilde g\in[g]\text{ with }\widetilde R=0\}.$$

Here $\overline B_1$ is the unit ball in $\mathbb R^n$, $g_{\mathbb R^n}$ is the Euclidean metric, and $\Theta_{\overline B_1,g_{\mathbb R^n}}$ is the corresponding Euclidean-ball value. Strict isoperimetric-ratio conjecture. If $(M,g)$ is not conformally diffeomorphic to $(\overline B_1,g_{\mathbb R^n})$, then

$$\Theta_{M,g}>\Theta_{\overline B_1,g_{\mathbb R^n}}.$$

This is the analogue of the strict inequality in the Yamabe problem. Together with the theorem that the supremum is achieved whenever it is strictly larger than the Euclidean-ball value, the conjecture would reduce the existence question to the conformal Euclidean ball case.