The disk maximizer conjecture for Neumann eigenvalue sums

Let $\Omega$ be a bounded convex plane domain of area $A$ and moment of inertia $I$, and let $\mu_2,\ldots,\mu_n$ be its positive Neumann Laplace eigenvalues. Neumann sum conjecture. For each $n\geq2$, the scale-invariant sum

$$\left.\left(\mu_2+\dots+\mu_n\right)\frac{A^3}{I}\right|_{\Omega}$$

should be maximal when $\Omega$ is a disk. The case $n=2$ is known from the Szegő–Weinberger inequality, while the asserted result for all sums and convex domains remains open in the paper.