  We consider a $J_b(\mathbb{Q}_p)$-torsor on the supersingular locus of the
Siegel threefold constructed by Caraiani-Scholze, and show that it induces an
isomorphism between a free group on a finite number of generators, and the
group of self-quasi-isogenies of a supersingular abelian surface, respecting a
principal polarization and a prime-to-$p$ level structure. Along the way, we
classify certain pro-\'etale torsors in terms of the pro-\'etale fundamental
group, describe the category of geometric covers of non-normal schemes, and use
this to compute pro-\'etale fundamental groups of curves.
