Multiplicator freeness for restricted-ramification Galois groups over global function fields

Let $K$ be a global function field with full constant field ${\mathbb{F}}_q$, let $T$ be a finite set of finite places, and let $\mathrm{\infty }$ be a distinguished place. We study the maximal pro-$\mathrm{\ell }$ extension of $K$ which is unramified outside $T$ and in which $\mathrm{\infty }$ splits completely. Using factorisation results for $\mathrm{\ell }$-primary idele class characters, we prove that its Galois group is multiplicator-free if $\mathrm{\ell }\nmid q-1$ or $T\mathrm{\ne }\mathrm{\varnothing }$. In particular, if $T\mathrm{\ne }\mathrm{\varnothing }$, the conclusion holds for every prime $\mathrm{\ell }$.