Schneider's exact-diameter conjecture for components of $D(k,q)$

Let $C\!D(k,q)$ be a connected component of $D(k,q)$. Schneider's conjecture. For every prime power $q$, $\operatorname{diam}(C\!D(3,q))=6$. For $k>3$ odd, the diameter is $k+5$, and for $k$ even, it is $k+4$, provided that $q$ is a sufficiently large prime power. Parts of this conjecture are known: the $k=3$ case for odd prime powers and certain $k=4$ cases were proved, while general cases remain open.