Free abelianity conjecture for fundamental groups of Galois covers

Let $X$ be a surface that can be degenerated to one of $U_{0,5,1}$, $U_{0,5,2}$, $U_{0,5,3}$, $U_{0,5,5}$, $U_{0,6,2}$, $U_{0,6,3}$, or $U_{3,5}$, and let $X_{\operatorname{Gal}}$ denote its Galois cover. The group $\pi_1(X_{\operatorname{Gal}})$ is the fundamental group of this cover.

Free abelianity conjecture. The group $\pi_1(X_{\operatorname{Gal}})$ is a free abelian group.

The conjecture arises because, in these cases, the fundamental group can be normally generated by one or two fork relations; based on results in smaller degrees, the authors expect the conjugates of a unique generator to commute. Determining the isomorphism classes of these groups remains an open question.