Lazebnik–Ustimenko–Woldar's girth conjecture for Wenger-type graphs

Let $D(k,q)$ be the graph defined by the parameters $k$ and the prime power $q$. Lazebnik–Ustimenko–Woldar's conjecture. For every prime power $q\ge 4$, the graph has girth

$$\operatorname{girth}(D(k,q))= \begin{cases} k+5, & k\text{ odd},\\ k+4, & k\text{ even}. \end{cases}$$

The source records matching lower bounds and presents equality as conjectural; no general resolution is supplied.