Finite-time round-point convergence conjecture for the torsion-driven flow

Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^n$. Let $X(t,x)$ be the flow defined by

$$\frac{\partial X}{\partial t}(t,x)=-\frac{1}{|\nabla u(t)(X(t,x))|^2}(X(t,x)\cdot\nu)\nu,\qquad X(0,x)=x,$$

where $\nu$ is the unit outer normal, $u(t)$ is the torsion function on the domain $\Omega_t$ enclosed by $X(t,\partial\Omega)$, and the flow is considered while defined. Finite-time round-point convergence conjecture. The convexity property is preserved throughout the evolution, and $X(t,\partial\Omega)$ converges to a round point in $C^\infty$ norm at a finite time $T$. The preceding proposition establishes monotonicity of the normalized torsional rigidity along the flow under the assumption that convexity is preserved. The conjecture asserts both preservation of convexity and finite-time asymptotic roundness, which are needed to complete the flow approach to the corresponding shape optimization problem.