Gaussian level-density conjecture for spin chains

Let $N$ be the number of particles, and let the level density mean the distribution of energy levels of the spin chain.

Gaussian level-density conjecture. For $N\gg1$, the level density follows a Gaussian distribution.

The conjecture concerns the asymptotic spectral statistics as the number of particles grows. The paper reports numerical support for Gaussian energy distributions, but does not establish the claim analytically.