Girth conjecture for D(k,q)

Prove that D(k,q) has girth k+5 for odd k and girth k+4 for even k, for all prime powers q≥4.

Background

The paper gives the general lower bounds girth(D(k,q))≥k+5 for odd k and ≥k+4 for even k. It states that equality is known only for certain infinite families and describes the conjecture as wide open.

References

Conjecture 6. The graph D(k, q) has girth k + 5 for odd k and girth k + 4 for even k, and all prime powers q ≥ 4. The conjecture is wide open, and it was confirmed only for a few infinite families of k and q, see Schliep [115], Thomason [124, 125], Xu, Cheng and Tang [152] and Xu [150].

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Conjecture 6, Section 6.3