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

Let $n>4$, let $B_1$ be the unit ball, and consider the boundary-value problem

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

Extremal-parameter conjecture. The extreme value for $P_{\lambda}$ is

$$\lambda^{*}=\frac{n^{3}(n-4)}{16}.$$

The preceding analysis gives at most two solutions for $0<\lambda<\frac{n^{3}(n-4)}{16}$, a unique solution at the proposed extremal value, and no solution above it. If the conjecture holds, it yields a hemisphere rigidity theorem under the stated lower bound on the $Q$-curvature, boundary agreement with the standard metric, and totally geodesic boundary; a geometric explanation for the extremal curvature bound remains open.