Beauville structures in $p$-central quotients

We prove a conjecture of Boston that if $p\geq 5$, all $p$-central quotients of the free group on two generators and of the free product of two cyclic groups of order $p$ are Beauville groups. In the case of the free product, we also determine Beauville structures in $p$-central quotients when $p=3$. As a consequence, we give an explicit infinite family of Beauville $3$-groups, which is different from the only one that was known up to date.