Localization-length conjecture for eigenvectors of random banded matrices

Let $A$ be a random banded matrix with independent entries and bandwidth $b$, and let $l$ denote its eigenvector localisation length. Localization-length conjecture. The eigenvector localisation length satisfies

$$l\asymp b^2.$$

This is the eigenvector analogue of the Poisson/Gaudin–Mehta conjecture. The supplied text does not state a resolution; the paper presents the relation as a conjecture concerning eigenvector statistics.