Vanishing of KAM islands in the thermodynamic limit

Let $V_N$ denote the relative phase-space volume of the union of KAM islands for an $N$-particle system at fixed energy density and fixed forcing parameters. Vanishing-of-islands conjecture.

$$V_N \to 0 \quad\text{as } N \to \infty.$$

This is suggested by analogy with the Fermi–Pasta–Ulam–Tsingou problem and the Boltzmann ergodic hypothesis, but the paper does not establish the limit or verify it for a concrete many-particle configuration.