Hang–Yang conjecture on positive critical points of the perturbed Paneitz problem

Let $S^3$ carry its standard metric $g_{S^3}$, let $P_{S^3}$ denote the Paneitz operator, and let $\epsilon>0$ be a small constant. Consider a positive smooth function $u_\epsilon$ satisfying

$$P_{S^3}u_\epsilon+\epsilon u_\epsilon=-u_\epsilon^{-7}\qquad\mathrm{on}\ S^3.$$

Hang–Yang conjecture. Every such solution $u_\epsilon$ is constant.

This is the assertion that all positive critical points of the perturbed Paneitz variational problem on $S^3$ are constant. The supplied passage gives no evidence that the conjecture has been resolved.