Ignat–Merlet conjecture on the viscosity solution as a line-energy minimizer

Let $\Omega\subset\mathbb{R}^2$ be a convex domain, let $\varphi_0(x)=\operatorname{dist}(x,\partial\Omega)$, and let $\overline{\mathcal{I}_f}$ be the relaxation of the line energy $\mathcal{I}_f$ in $L^1$. Assume that $\overline{\mathcal{I}_f}$ is lower semicontinuous in $L^1$.

Ignat–Merlet conjecture. Then $(\nabla\varphi_0)^\perp$ is a global minimizer of $\mathcal{I}_f$.

Here $(\nabla\varphi_0)^\perp$ is the viscosity solution associated with the distance function in the regular-domain setting. The parser marks this candidate as resolved, so it is recorded as solved.