Grigorchuk-Gupta-Sidki groups as a source for Beauville surfaces

If $G$ is a Grigorchuk-Gupta-Sidki group defined over a $p$-adic tree, where $p$ is an odd prime, we study the existence of Beauville surfaces associated to the quotients of $G$ by its level stabilizers ${\text{st}}_G(n)$. We prove that if $G$ is periodic then the quotients $G\mathrm{/}{\text{st}}_G(n)$ are Beauville groups for every $n\ge 2$ if $p\ge 5$ and $n\ge 3$ if $p=3$. On the other hand, if $G$ is non-periodic, then none of the quotients $G\mathrm{/}{\text{st}}_G(n)$ are Beauville groups.