Cyclotomic and modified cyclotomic difference-set classification conjecture

Let $q$ be a prime power, let $m$ divide $q-1$, and let $H_{q,m}$ and $M_{q,m}$ denote the $m$th-cyclotomic and modified cyclotomic subsets of the finite field $\mathbb{F}_q$, respectively. A difference set is called nontrivial when it is neither empty nor the whole group.

Cyclotomic difference-set classification conjecture. $H_{q,m}$ is a nontrivial difference set in $\mathbb{F}_q$ if and only if one of the following holds: (a) $m=2$ and $q\equiv3\pmod{4}$; (b) $m=4$ and $q=p=1+4t^2$ for some odd integer $t$; or (c) $m=8$ and $q=p=1+8u^2=9+64v^2$ for some odd integers $u$ and $v$. Moreover, $M_{q,m}$ is a nontrivial difference set in $\mathbb{F}_q$ if and only if one of the following holds: (a') $m=2$ and $q\equiv3\pmod{4}$; (b') $m=3$ and $q=16$; (c') $m=4$ and $q=p=9+4t^2$ for some odd integer $t$; or (d') $m=8$ and $q=p=49+8u^2=441+64v^2$ for some odd integer $u$ and even integer $v$.

The conjecture is known for odd $m$ and for even $m\leqslant22$; the unresolved cases concern the remaining parameters, particularly even $m>22$ other than $2$, $4$, and $8$.