Maximum-curvature conjecture for optimal control of moving sets

Let $\Omega(t)$ be the evolving set with boundary parametrized by $x(t,\xi)$, and let the optimal control be the normal velocity control acting along this boundary. Denote the curvature of the boundary at $x(t,\xi)$ by $\omega(t,\xi)$. Maximum-curvature conjecture. At every time $t\in[0,T]$, the optimal control is active precisely along the portion of the boundary where the curvature is maximum. This claim describes the geometric localization of the optimal control suggested by the backward dual equation: characteristics with larger curvature cause the dual variable to grow faster backward in time, making those boundary portions the most advantageous locations for shrinking the set. The parser provides no evidence that the claim has been proved or disproved.