Geometric properties of optimizers for the maximum gradient of the torsion function

Consider $J(\Omega):= \|\nabla u_\Omega\|_\infty/\sqrt{|\Omega|}$ and $J_P(\Omega):= \|\nabla u_\Omega\|_\infty/P(\Omega)$, where $\Omega$ is a planar convex domain, $u_\Omega$ is the torsion function, $P(\Omega)$ is the perimeter of $\Omega$ and $|\Omega|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_\Omega|$ attains its maximum.