The minimal ramification conjecture for finite groups over finite fields

Let $G$ be a nontrivial finite group. Write $d(G)$ for the minimal number of generators of $G$, $G^{\mathrm{ab}}$ for its abelianization, and $p(G^{\mathrm{ab}})$ for the subgroup generated by the $p$-Sylow subgroups of $G^{\mathrm{ab}}$. An extension of function fields is geometric if the two fields have the same field of constants. Minimal ramification conjecture. There exists a geometric Galois extension $K/\mathbb{F}_q(T)$ with

$$\operatorname{Gal}(K/\mathbb{F}_q(T))=G$$

and ramified over at most $r$ prime divisors of $\mathbb{F}_q(T)$ if and only if

$$r\ge d\left(G^{\mathrm{ab}}/p(G^{\mathrm{ab}})\right).$$

This conjecture generalizes a corresponding conjecture of Boston and Markin over $\mathbb{Q}$ and concerns the minimum number of ramified prime divisors in a finite-group extension. The supplied status evidence says that significant progress is known for the symmetric and alternating groups, but the conjecture remains open for many values of $n$ and $q$.