The non-power word surjectivity conjecture for PSL(2,q)

Let $F_2$ be the free group on two generators, and let $w=w(x,y)\in F_2$. Call $w$ a power word if there are $v\in F_2$ and $m\in\mathbb{N}$ such that $w(x,y)=v(x,y)^m$.

Non-power word surjectivity conjecture. If $w$ is not a power word, then, for $G=\operatorname{PSL}(2,q)$,

$$w(G)=G.$$

This is presented as a variant of Shalev's conjecture and is motivated by the paper's trace-map questions. It remains conjectural in the source; the claim concerns all non-power two-variable words on the family $\operatorname{PSL}(2,q)$.