Isoperimetric inequality for fundamental tones of free plates

Let $\Omega\subset\mathbb{R}^d$ be a smoothly bounded region, and let $\Omega^*$ be the ball in $\mathbb{R}^d$ with the same volume as $\Omega$. For fixed $\tau>0$ and $\sigma\in(-1/(d-1),1)$, let $\omega_1(\Omega)$ denote the lowest nonzero eigenvalue of the free plate associated with the Rayleigh quotient

$$Q_\Omega[u]=\frac{\int_\Omega (1-\sigma)|D^2u|^2+\sigma(\Delta u)^2+\tau|Du|^2\,dx}{\int_\Omega u^2\,dx}.$$

Free-plate isoperimetric conjecture. One has

$$\omega_1(\Omega)\leq\omega_1(\Omega^*),$$

with equality if and only if $\Omega=\Omega^*$.

This conjecture extends the known zero-Poisson-ratio result to free plates with nonzero Poisson's ratio. The paper proves the inequality when the fundamental mode of the ball has simple angular dependence, but the general assertion remains open.