Boston–Markin conjecture over the rational function field

Let $G$ be a nontrivial finite group, let $d_\lhd(G)$ be the least number of conjugacy classes in $G$ that generate $G$, and let $q$ be a prime power coprime to $|G|$. A regular $G$-extension $K/\mathbb{F}_q(T)$ is a Galois extension with constant field exactly $\mathbb{F}_q$, and split completely at infinity means that the place at infinity splits completely in $K$.

Function-field Boston–Markin conjecture. There exists a regular $G$-extension $K/\mathbb{F}_q(T)$, split completely at infinity, such that the number of ramified primes of $\mathbb{F}_q(T)$ is $d_\lhd(G)$.

This is the function-field analogue of the tamely ramified Boston–Markin conjecture, with regularity excluding constant extensions and the coprimality condition ensuring tame ramification. The source identifies it as a special case of conjectures in the cited literature; the statement remains presented as conjectural.