Disk minimality conjecture for sums of Dirichlet eigenvalues

Let $\Omega$ be a bounded plane domain of diameter $D$, and let $0<\lambda_1\leq\lambda_2\leq\cdots$ be the Dirichlet eigenvalues of its Laplacian. Disk eigenvalue-sum conjecture. For every $n\geq1$, the quantity $(\lambda_1+\cdots+\lambda_n)D^2$ is minimal when $\Omega$ is a disk. The case $n=1$ follows from the Faber--Krahn and isodiametric theorems, while the general assertion remains open.