Stable anisotropic equilibria conjecture

Let $n\geq 2$, let $\rho^n$ denote the function associated with equilibria whose largest eigenspace has dimension $d=1$, and let $\rho^*>0$ be the threshold density. For $\rho\in(\rho^*,\infty)$, consider the solutions of

$$\rho^n(\lambda)=\rho.$$

The branch with largest $\lambda$ corresponds to the unique class of stable anisotropic equilibria.

This conjecture concerns the stability of the non-uniform equilibria in the kinetic model. It has been verified in dimensions $n=2$, $n=3$, and $n=4$, while stability in general dimension remains open.