Girao's weighted Alexandrov inequality conjecture for strictly convex hypersurfaces

Let $\Sigma\subset\mathbb S^n$ be a closed, orientable, connected embedded hypersurface, and let $H$ denote its mean curvature. For $x\in\mathbb S^n$, define

$$\rho_x(q)=\cos\left(d_{\mathbb S^n}(x,q)\right),$$

where $d_{\mathbb S^n}(x,q)$ is the geodesic distance from $q$ to $x$, and let $|\Sigma|$ be the area of $\Sigma$ and $\omega_{n-1}$ the area of the unit $(n-1)$-sphere. Girao's conjecture. If $\Sigma$ is strictly convex, then

$$\sup_{x\in\mathbb S^n}\int_\Sigma \rho_x H\,d\Sigma\ge (n-1)\omega_{n-1}\left[\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}}-\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n}{n-1}}\right].$$

Equality should hold if and only if $\Sigma$ is a geodesic sphere. This is an Alexandrov-type weighted curvature inequality in the sphere; the supplied source relates it to work of Girao, but gives no resolution status, so it remains open.