The degree-zero word reformulation for $\operatorname{PSL}_2$

Let $K$ be an algebraically closed field of characteristic $0$, let $w(x_1,\dots,x_m)$ be a group word whose degree in each variable $x_i$ is $0$, and set

$$G=\operatorname{GL}_2(K)/\{\pm 1\}.$$

Degree-zero word reformulation. The image of $w$ on $G$ must be

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

This is presented in the source as a reformulation of the word-image conjecture after reducing to words of degree zero in every variable. It remains open in the stated generality.