The exact hexagon-count conjecture for strongly regular graphs with parameters $\lambda=1$ and $\mu=2$

Let $G$ be a strongly regular graph with parameters $\lambda=1$ and $\mu=2$, order $n$, and valency $k$. Let $p_6$ denote the number of hexagons in $G$. Exact hexagon-count conjecture. The number of hexagons in $G$ is equal to

$$p_6=\frac{1}{12}nk(k-2)(2k^2-21k+53).$$

The paper proves the same expression as a lower bound. Equality holds when the auxiliary quantity $n_3$ is zero; this condition means that two triangles connected through two edges are necessarily connected through the third edge. The conjecture is motivated by the claim that otherwise many symmetries are broken. Under this equality condition, Makhnev proved that $srg(99,14,1,2)$ does not exist.