Energy scale below which ETH matrix elements become random-matrix-like
In plain words
ETH says the table of a local observable between energy eigenstates looks random, but only for eigenstates very close in energy. How wide that energy window is, and how it shrinks with system size, is not settled.
Precise statement
For a chaotic local Hamiltonian on $L$ sites and a local observable $O$, define $E_{\mathrm{RMT}}(L)$ as the largest energy difference $\omega$ such that off-diagonal elements $O_{mn}$ with $\left|E_m - E_n\right| < \omega$ are statistically invariant under unitary rotations, including the higher-order correlations of full ETH in its free-probability form. Determine $E_{\mathrm{RMT}}(L)$ for systems with and without diffusive conserved quantities; with diffusion the natural guess is $E_{\mathrm{RMT}} \sim \hbar D/L^{2}$. Answer: a scaling form established on at least one chaotic spin chain.
What would settle it
Computation of higher-order matrix-element correlations (free cumulants) in chaotic chains up to $L \sim 22-24$ by full diagonalization, with typicality methods for larger $L$, and a size scaling of the onset energy.
Status in the literature
The free-probability formulation of full ETH (Pappalardi, Foini, Kurchan 2022) supplies the criterion; the size scaling is numerically unsettled.