Low-frequency spectral asymptotics for the Poincaré operator on ellipsoids

Let $\boldsymbol{ abla}$ be the relevant geometric setting on the unit cosphere bundle $U^\star S^2$, let $k_\omega$ be the Hamiltonian of the Kelvin equation, and let $n(x,\boldsymbol{\xi})$ denote the number of values of $\omega$ such that $k_\omega(x,\boldsymbol{\xi})=0$. For an ellipsoid $E$, let $\mu_{j,n}$ be the eigenvalues of the Poincaré operator with polynomial eigenvectors of degree $n$, and let $H_E$ be the Hamiltonian of the Kelvin equation. For $0<a,b<\omega_-$, the conjecture is

Low-frequency spectral asymptotics.

$$\#\{\mu_{j,n}\in[a,b]\}\sim \frac{2n+3}{8\pi^2}\int_{U^\star S^2\cap\{(x,\boldsymbol{\xi})\mid k_\omega(x,\boldsymbol{\xi})=0,\ \omega\in[a,b]\}} n\,dL$$

when $n\to+\infty$, where $L$ is the Liouville measure on $U^\star S^2$. This would give the asymptotic distribution of the low-frequency eigenvalues in $[0,\omega_-[$ and complement the Weyl asymptotic formula known in the interval $[\omega_-,\omega_+]$.