The genus-zero covering subgroup conjecture

Let $U_i$ be the genus-zero hyperbolic curves and let $\Pi_{U_i}^{(2,\ell)}$ be the groups used in the source for a prime $\ell$. Let $(\overline{\Pi}_{U_i}^{2,\ell})^{[1]}$ denote the indicated commutator subgroup, and let $(\Pi_{U_i}^{(2,\ell)})_{g=0}$ be the intersection of all abelian open subgroups corresponding to coverings of genus zero. The genus-zero covering subgroup conjecture. For every prime $\ell$,

$$(\Pi_{U_i}^{(2,\ell)})_{g=0}\subset(\overline{\Pi}_{U_i}^{2,\ell})^{[1]}.$$

This is posed as a reformulation of the question whether genus-zero hyperbolic curves have enough coverings that also have genus zero. The supplied context does not state a resolution.