Global bi-Lipschitz regularity conjecture for minimisers between doubly connected domains

Let $D$ and $\Omega$ be doubly connected domains with smooth boundaries, and let $\rho$ be a smooth non-vanishing metric defined on the closure of $\Omega$. Write $\operatorname{Mod}(D)$ and $\operatorname{Mod}(\Omega)$ for their moduli. Global bi-Lipschitz regularity conjecture. If

$$\operatorname{Mod}(D)\leq \operatorname{Mod}(\Omega),$$

then the minimiser of the $\rho$-energy is globally bi-Lipschitz continuous and has a smooth extension up to the boundary. The conjecture is motivated by the critical J. C. C. Nitsche configuration, where the minimiser need not be bi-Lipschitz when the modulus inequality fails; the cited context notes that the corresponding Nitsche conjecture for annuli was solved, while this general metric-valued boundary regularity statement remains unresolved.