Hyperbolic Weinstock inequality for convex domains

Let $\Omega$ be a bounded convex domain in $\mathbb{H}^n$. Let $\Omega^*$ be a geodesic ball with $|partialOmega^*|=|partialOmega|$.

Hyperbolic Weinstock conjecture. Is it true that

$$\sigma_1(\Omega)\leqsigma_1(\Omega^*)$$

and equality holds if and only if $\Omega$ is a geodesic ball?

This asks whether geodesic balls maximize the first Steklov eigenvalue among bounded convex domains in hyperbolic space with fixed boundary measure. The corresponding Euclidean inequality is known for convex domains, but for higher dimensions the hyperbolic statement remains open to the authors' knowledge.