The degree-zero word-map conjecture on GL2 modulo its center

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 $G={\rm GL}_2(K)/\{\pm 1\}$. Degree-zero word-map conjecture. The image of $w$ on $G$ is ${\rm PSL}_2(K)$. This is presented as a reformulation of the surjectivity conjecture for ${\rm PSL}_2$; the source gives no resolution.