The wreath-product 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 $p\leq q\leq r$, and choose an integer $m$ such that the alternating group ${\cal A}_m$ can be generated by two $p$-cycles. Let ${\cal A}_{\mathrm{deg}}$ be the alternating group on $\mathrm{deg}$ letters and ${\cal A}_m\wr {\cal A}_{\mathrm{deg}}$ the corresponding wreath product. Wreath-product quotient conjecture. For all but finitely many integers $\mathrm{deg}$, $G$ maps onto

$${\cal A}_m\wr {\cal A}_{\mathrm{deg}}.$$

This conjecture extends the preceding quotient construction from the semidirect product $C_p^{\mathrm{deg}-1}\rtimes {\cal A}_{\mathrm{deg}}$ to wreath products. The source does not state whether the conjecture has been resolved.