Polar-dual Neumann eigenvalue sum conjecture for convex domains

Let $\Omega \subset \mathbb{R}^d$ be a bounded convex domain containing the origin. Write $\mu_j$ for its Neumann Laplace eigenvalues, $V$ for volume, $I$ for the moment of inertia, and $\Omega^\circ$ for the polar dual of $\Omega$, defined by

$$\Omega^\circ=\{x\in\mathbb{R}^d:x\cdot y<1\text{ for all }y\in\overline{\Omega}\}.$$

Polar-dual Neumann eigenvalue sum conjecture. The scale-invariant quantity

$$\left.(\mu_2+\dots+\mu_n)V^{2/d}\right|_\Omega\left.\frac{V^{1+2/d}}{I}\right|_{\Omega^\circ}$$

is maximal when $\Omega$ is a ball, for each $n\geq2$. This conjecture proposes the maximizer for the Neumann analogue of a bounded polar-dual eigenvalue expression; the corresponding Dirichlet quantity is bounded above using John’s ellipsoid theorem, but the maximizing statement for general convex domains is presented as an open problem.