The Schmidt–White conjecture on cyclotomic strongly regular graphs

Let $q=p^{\ell}$ be a prime power, let $m$ be a positive integer, and let $N$ divide $(q^m-1)/(q-1)$. Let $C_0^{(N,q^m)}$ be the subgroup of $\mathbb{F}_{q^m}^*$ of index $N$, and let $\operatorname{Cay}(\mathbb{F}_{q^m},C_0^{(N,q^m)})$ denote the Cayley graph with connection set $C_0^{(N,q^m)}$. A graph is strongly regular if it is regular and every pair of adjacent vertices and every pair of nonadjacent vertices have respectively constant numbers of common neighbors. The Schmidt–White conjecture. Assume that $N\mid(q^m-1)/(q-1)$. Then $\operatorname{Cay}(\mathbb{F}_{q^m},C_0^{(N,q^m)})$ is strongly regular if and only if it is either a subfield example, a semi-primitive example, or one of the eleven sporadic examples in Table 1 of the source. This conjecture aims to classify the cyclotomic strongly regular graphs; the source states that it remains open, while the stated necessary-and-sufficient conditions provide a way to search for further examples.