The Suzuki-group construction has edge-girth parameter q-1

Let $q=2^{2e+1}$, and let $\mathcal G_q(V,E)$ be the graph constructed from the biaffine-plane and Suzuki-group construction. An $egr(v,k,g,\lambda)$-graph is an edge-girth-regular graph with order $v$, degree $k$, girth $g$, and each edge lying in exactly $\lambda$ cycles of length $g$.

The edge-girth conjecture. The family of graphs $\mathcal G_q(V,E)$ is a family of

$$egr(q(q^2+1),q,5,q-1)$$

-graphs.

The construction is already known to give graphs with the displayed order, degree, and girth, while the conjecture asserts that every edge lies in exactly $q-1$ pentagons. It was checked computationally for $q=8$; the general case remains open.