Symmetry conjecture for the Robin torsion overdetermined system

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, let $\nu$ denote the unit outward normal on $\partial\Omega$, and let $\beta>0$. Consider the overdetermined system

$$\begin{cases} -\Delta u=1 & \text{in }\Omega,\\ \frac{\partial u}{\partial\nu}+\beta u=0 & \text{on }\partial\Omega,\\ -\beta^2u^2+\frac{1}{2}|\nabla u|^2+\frac{\beta}{2}u^2H-u=\text{constant} & \text{on }\partial\Omega, \end{cases}$$

where $H$ is the mean curvature of $\partial\Omega$. Robin torsion symmetry conjecture. For any $\beta>0$, the domain for which this overdetermined system admits a solution must be a ball. This would extend the classical symmetry conclusion in Serrin's Dirichlet limit and characterize balls among domains supporting the critical overdetermined system associated with the Robin torsion functional. The ball is known to be the unique volume-constrained minimizer of that functional, but the asserted classification of all domains admitting a solution remains open.