Ustimenko's spectral bound conjecture for components of $D(k,q)$

Let $C\!D(k,q)$ be a connected component of $D(k,q)$, and let $\lambda_2$ denote its second largest eigenvalue. Ustimenko's conjecture. For all $k$ and $q$,

$$\lambda_2\le 2\sqrt q.$$

This is a universal spectral bound for the components; the source gives no resolution.