The alternating-group quotient conjecture for triangle groups

Let $G=\Delta(p,q,r)$ be the triangle group with presentation

$$\Delta(p,q,r)=\langle x,y\mid x^p=y^q=(xy)^r=1\rangle.$$

Assume that $p$ is prime and $p\leq q\leq r$. Write $C_p$ for the cyclic group of order $p$, and let ${\cal A}_{\mathrm{deg}}$ be the alternating group on $\mathrm{deg}$ letters. Alternating-group quotient conjecture. For all but finitely many integers $\mathrm{deg}$, $G$ maps onto

$$C_p^{\mathrm{deg}-1}\rtimes {\cal A}_{\mathrm{deg}}.$$

This is one of two conjectures suggested by the paper's method for composing permutation representations, extending the construction of alternating-group quotients of triangle groups. The source does not state whether the conjecture has been resolved.