Pucci–Serrin conjecture for radial solutions of polyharmonic equations

Let $B$ be the unit ball of $\mathbb{R}^n$, and let $k\in\mathbb{N}$ satisfy $n>2k\geq 2$. Consider $\lambda\in\mathbb{R}$ and $u\in C^{2k}(\overline{B})$ solving

$$\left\{\begin{array}{cc} \Delta^k u-\lambda u=|u|^{2^\star-2}u&\text{ in }B\\ u=\partial_\nu u=...=\partial_{\nu}^{k-1}u=0&\text{ on }\partial B, \end{array}\right.$$

where $2^\star:=\frac{2n}{n-2k}$. Pucci–Serrin conjecture. If $2k<n<4k$, then there exists $\lambda_0(n,k)>0$ such that for every $0<\lambda<\lambda_0(n,k)$, every radial solution is identically zero. The paper's abstract says that this conjecture is proved under a uniform bound on the energy, while the supplied context does not establish whether the full conjecture is resolved; this status should therefore be checked against the paper's results.