The perturbation-limit conjecture for the minimal surface boundary value problem

Let $\Omega$ be a convex domain and let $\alpha\in(0,\pi)$. Let $u$ be the solution of the boundary value problem BVP 3.1, and for each $\varepsilon>0$ let $w_\varepsilon$ be a solution of the perturbation problem BVP 3.3.

Perturbation-limit conjecture. Up to addition of a constant, the smooth solution $u$ is the suitable limit of the sequence of solutions $w_\varepsilon$ as $\varepsilon\to0$.

This conjecture concerns whether the unperturbed minimal-surface boundary value problem can be obtained from the regularized problems by letting the perturbation parameter tend to zero. The source provides no resolution, so the conjecture remains open.