Liouville-type conjecture for the critical p-Laplacian on a half-space

Let $1<p<N$, let $p^*=Np/(N-p)$, and let ${\mathbb R}_+^N$ denote the half-space. A function $u\in \mathcal D^{1,p}_0({\mathbb R}_+^N)$ is a weak solution of the critical $p$-Laplace equation if

$$-\Delta_p u=|u|^{p^*-2}u\quad\textrm{in }{\mathbb R}_+^N.$$

Liouville-type conjecture. Every weak solution $u\in\mathcal D^{1,p}_0({\mathbb R}_+^N)$ of this equation is identically zero:

$$u\equiv 0.$$

The conjecture would rule out nontrivial sign-changing limiting profiles supported on a half-space in the global compactness analysis of Palais–Smale sequences for critical $p$-Laplacian problems, yielding a complete generalisation of Struwe's result to the $p$-Laplacian. The supplied text gives no resolution, so its status remains open.