Lazebnik–Ustimenko–Woldar's diameter upper-bound conjecture

Let $C\!D(k,q)$ be a connected component of $D(k,q)$, with $k\ge 2$ and $q$ a prime power. Lazebnik–Ustimenko–Woldar's conjecture. There is a positive constant $C$ such that

$$\operatorname{diam}(C\!D(k,q))\le (\log_{q-1}q)k+C.$$

This predicts a linear diameter bound with coefficient $\log_{q-1}q$; the source gives no resolution.