Uniqueness and least-gradient identification for the global median value problem

Let $\Omega\subset\mathbb{R}^{2}$ be strictly convex and let $g:\partial\Omega\to\mathbb{R}$ be continuous. Let $u^{*}$ denote the function of least gradient on $\Omega$ with boundary values $g$, and consider continuous solutions of the global median value property

$$u(x)=\operatorname*{median}_{s\in\partial B(x,r)}\{u(s)\}\quad\text{whenever }x\in\Omega\text{ and }B(x,r)\Subset\Omega.$$

Global median value conjecture for strictly convex domains. There exists a unique continuous solution $u$ with $u=g$ on $\partial\Omega$, and this solution satisfies $u=u^{*}$. This proposed result would establish both well-posedness of the global Dirichlet problem in strictly convex domains and its equivalence with the least-gradient construction; the supplied text gives no resolution evidence, so it remains open.