Faithful Galois action on unmarked triangle curves

A triangle curve is a smooth curve $D$ equipped with an effective action of a finite group $G$ such that $D/G \cong \mathbb{P}^{1}_{\mathbb{C}}$ and the quotient map is branched over $\{0,1,\infty\}$. An unmarked triangle curve means that the group action is considered up to isomorphism without fixing an embedding of $G$ into $\operatorname{Aut}(D)$.

Triangle-curve faithfulness conjecture. The absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of isomorphism classes of unmarked triangle curves.

This is identified in the text as the crucial property needed to obtain the desired Galois action on moduli components; no proof or resolution is supplied.