Disk minimization conjecture for convex-domain Neumann eigenvalue sums

Let $\Omega$ be a convex domain of given diameter $D$, and let $0=\mu_1(\Omega)<\mu_2(\Omega)\leq\mu_3(\Omega)\leq\cdots$ be the Neumann Laplacian eigenvalues. Disk minimization conjecture. For each integer $n\geq 3$, the scale-invariant sum

$$\bigl(\mu_2(\Omega)+\dots+\mu_n(\Omega)\bigr)D^2$$

is minimized when $\Omega$ is a disk. This is suggested by the proved result for triangular domains and the proposition for ellipses, but the assertion for general convex domains remains open.