Uniqueness and characterization of optimal measures for negatively curved shells

Let $\Omega$ be a simply connected shell, let $p$ denote its prescribed elevation, and write $d_{\partial\Omega}$ for the distance to $\partial\Omega$. Let $M$ be the medial axis of $\Omega$, and let $\varrho$ be the change-of-measure factor from the Lebesgue disintegration along paths of quickest exit. Assume

$$\det\nabla\nabla p\leq 0$$

a.e. and that the paths of quickest exit from $\Omega$ do not meet at $\partial\Omega$.

Uniqueness conjecture. The optimal measures $\mu$ are unique and satisfy

$$\mu=\lambda\nabla^{\perp}d_{\partial\Omega}\otimes\nabla^{\perp}d_{\partial\Omega}\,dx\quad\text{on }\Omega,$$

where

$$-\frac{1}{2\varrho}\partial_{\nabla d_{\partial\Omega}}^{2}\left(\varrho\lambda\right)=\det\nabla\nabla p \quad\text{on }\Omega\backslash\overline{M},$$

and

$$\varrho\lambda=\partial_{\nabla d_{\partial\Omega}}\left(\varrho\lambda\right)=0 \quad\text{at }\overline{M}.$$

This conjecture extends the observed uniqueness of optimal $\mu$ beyond the convex examples to simply connected, negatively curved shells, provided the quickest-exit paths do not meet at the boundary. The source gives no resolution of the conjecture.