Higher-girth edge-girth-regular graph families

Let $q\geq 3$ be a prime power, and let $g\in\{8,12\}$. 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 $g$.

Higher-girth existence conjecture. For $q\geq 3$ a prime power and $g\in\{8,12\}$, there exists a family of

$$egr\left(2q^{\frac{g-2}{2}},q,g,(q-1)^{\frac{g-2}{2}}(q-2)\right)$$

-graphs.

The conjecture is motivated by incidence graphs of generalized quadrangles and hexagons and by Moore-cage constructions. It predicts families with the indicated order and edge-girth parameters; no general construction or proof is supplied in the paper.