Restricted inverse Galois problem over function fields

Let $p$ be a prime and let $G$ be a nontrivial finite group. Write $G^{ab}$ for the abelianization of $G$, let $p(G)$ be the normal subgroup generated by the elements of $p$-power order, and let

$$d=d\left((G/p(G))^{ab}\right).$$

For the global function field $\mathbb{F}_p(t)$, let $\operatorname{Ram}_{\mathbb{F}_p(t)}(G)$ denote the minimum number of places ramified in a $G$-extension of $\mathbb{F}_p(t)$. Restricted inverse Galois conjecture. There exists a $G$-extension of $\mathbb{F}_p(t)$ such that

$$\operatorname{Ram}_{\mathbb{F}_p(t)}(G)=\begin{cases} d+1 & \text{if } p\mid |G^{ab}|,\\ \max(d,1) & \text{if } p\nmid |G^{ab}|. \end{cases}$$

This refines the problem of minimizing ramification in geometric Galois extensions of rational function fields, combining the expected contribution from the abelianization with the behavior of the normal subgroup generated by elements of $p$-power order. The source presents the statement as a conjecture based on related work and families of examples; its resolution status is not specified here.