The word-image conjecture for the projective linear group

Let $K$ be an algebraically closed field of characteristic $0$, let $w(x_1,\dots,x_m)$ be a nontrivial group word, and let $\operatorname{PSL}_2(K)$ denote the projective linear group. Word-image conjecture. The image of $w$ on $\operatorname{PSL}_2(K)$ is

$$\operatorname{PSL}_2(K).$$

This conjecture asks whether every nontrivial word map is surjective on $\operatorname{PSL}_2(K)$. The image is known to contain all non-unipotent elements, while the question of whether the nontrivial unipotent class lies in the image remains open.