Higher-dimensional convergence conjecture for the transformed solutions

Let $cOmega$ be the domain and $comega\subset\Omega$ the region appearing in the problem, and let $u_n$ be the solution of the problem denoted by (pbn), given by Theorem 1 with $M(x)\equiv I$. Define

$$v_n=\frac{u_n^{n+1}}{n+1}.$$

Higher-dimensional convergence conjecture. The sequence $cv_n$ is bounded in $W_0^{1,2}(\Omega)\cap L^\infty(\Omega)$ and therefore converges, up to subsequences, to a bounded nonnegative function $v$. Moreover, $v$ is a weak solution of

$$\begin{cases} -\Delta v+\dfrac{|\nabla v|^2}{v}=f & \text{in $\omega$,} \\ v=0 & \text{on $\partial\omega$,} \end{cases}$$

and $v\equiv0$ in $cOmega\setminus\omega$. This conjecture concerns the higher-dimensional extension of the limiting result for the approximating elliptic problems; the source states it as an open problem and provides no resolution.