Poisson/Gaudin–Mehta conjecture for random symmetric banded matrices

Let $A$ be a random real symmetric banded matrix of size $N\times N$ with independent entries and bandwidth $b\asymp N^\alpha$. Poisson/Gaudin–Mehta conjecture. The limiting local statistics of $A$ are Poisson if $\alpha<\frac{1}{2}$ and Gaudin–Mehta if $\alpha>\frac{1}{2}$. This is a central open problem in random matrix theory. Existing results and numerical and heuristic evidence support the critical bandwidth scale $b\asymp\sqrt{N}$, but do not establish the claimed local statistics in the two regimes.