The curvature bound conjecture for adaptive anisotropic total variation eigenfunctions

Let $C\subset\mathbb{R}^2$ be a bounded ellipse whose indicator function is an eigenfunction in the sense of the adaptive anisotropic total variation eigenfunction equation, and let $\kappa(x)$ denote the curvature at $x\in\partial^*C$. Here $P(C)$ and $|C|$ denote the perimeter and area of $C$, respectively, and $\lambda_C^A$ is the corresponding adaptive anisotropic total variation eigenvalue. Curvature bound conjecture.

$$\max_{\forall x\in\partial^*C}\kappa(x)\leq\frac{\lambda_C^A}{a^4}=\frac{1}{a^3}\frac{P(C)}{|C|}.$$

The conjecture records the numerically observed $a^{-3}$ dependence of the critical bound for ellipses, relating the maximal curvature to the adaptive anisotropic total variation eigenvalue and the perimeter-to-area ratio. The supplied text gives numerical evidence but no proof or resolution, and explicitly notes that decoupling curvature from convexity and other shape structures is difficult.