Equilateral triangle conjecture for torsional rigidity under minimal width

Let $\Omega$ be a convex domain in $\mathbb{R}^2$ with unit minimal width. Let $u\in H_0^1(\Omega)$ solve

$$-\Delta u=1.$$

Its torsional rigidity is

$$T(\Omega)=\int_\Omega |\nabla u|^2.$$

Equilateral triangle torsional-rigidity conjecture. The equilateral triangle of unit width minimizes the torsional rigidity among shapes having unit minimal width.

The conjecture is motivated by comparison with classical optimization problems for Dirichlet-Laplace eigenvalues and torsional rigidity, together with numerical simulations. Its resolution is not supplied here.