Lipschitz continuity conjecture for Riemannian max-flow solution sets

Let $g_{ij}(\tau)$ be a continuous family of metrics on a Riemannian manifold $\left(M^d,\partial\right)$ whose boundary has non-outward-pointing mean curvature. Let $C_\tau\subset\partial M$ be a $d-2$ manifold dividing some number of boundary components of $M$ into pieces $A$ and $B$, with $A\cap B=C$. Fix a class

$$x\in H_{d-1}(M,C;Z)/\operatorname{torsion}$$

and consider the max-flow problem for flows dual to $x$. Let $K_\tau$ be the infinite-dimensional convex set of max flows. Lipschitz continuity conjecture. The set $K_\tau$ varies in a Lipschitz continuous fashion as a function of $\tau$, viewed as a subset of the Banach space of vector fields with the pointwise ($L^\infty$) norm. The conjecture is the Riemannian analogue of the established continuity result for perturbed graph max-flow problems. It remains open because unresolved analytical questions prevent the conjecture from being stated as a theorem.