Extremal-parameter conjecture for the exponential biharmonic problem $Q_{\lambda}$

Let $B_1$ be the unit ball, and consider the boundary-value problem

$$\begin{cases} \Delta^{2}u=\lambda e^{4u} & \text{in } B_{1},\\ u=1 & \text{on } \partial B_{1},\\ \frac{\partial u}{\partial n}=-1 & \text{on } \partial B_{1}. \end{cases}$$

Extremal-parameter conjecture. For the problem $Q_{\lambda}$, the extremal parameter is

$$\lambda^{*}=8.$$

The claim concerns the threshold parameter for solvability of this exponential biharmonic boundary-value problem. The supplied text gives the asserted value but no resolution or further context, so its status remains open.