Herman's conjecture on KAM stability of Diophantine elliptic equilibria

Let $H$ be a real-analytic Hamiltonian with an elliptic equilibrium at the origin, and let $\alpha\in\mathbb R^n$ be the frequency vector of the linear part of the flow. Recall that $\alpha$ is Diophantine if there exist $\gamma>0$ and $\tau\geq n-1$ such that

$$|k\cdot\alpha|\geq\gamma|k|_1^{-\tau}$$

for every $k=(k_1,\dots,k_n)\in\mathbb Z^n\setminus\{0\}$, where $|k|_1=|k_1|+\cdots+|k_n|$. Herman's conjecture. If $\alpha$ is Diophantine, then every sufficiently small neighborhood of the origin contains a set of positive Lebesgue measure consisting of Lagrangian invariant tori. This asserts KAM stability without any non-degeneracy assumption. It is known for $n=2$, even in the smooth category, but remains unknown in general.