Optimal concavity of the torsion function

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\ge 2$ and $\mathrm{\Omega }$ be a bounded open set in ${\mathbb{R}}^n$ whose torsion function $u$ (i.e. the solution to $\mathrm{\Delta }u=-1$ in $\mathrm{\Omega }$, vanishing on $\mathrm{\partial }\mathrm{\Omega }$) satisfies the following property: $\sqrt{M-u(x)}$ is convex, where $M=\max \{u(x):x\in \bar{\mathrm{\Omega }}\}$. Then $\mathrm{\Omega }$ is an ellipsoid.