Surjectivity conjecture for word maps on $PSL_2(\mathbb C)$

Let $F(x,y)$ be the free group on two generators, and let $G=PSL_2(\mathbb C)$. A word map is the map induced by a word $w=w(x,y)\in F(x,y)$,

$$w\colon G\times G\to G.$$

Surjectivity conjecture. For every non-identity word $w\in F(x,y)$, the word map

$$w\colon PSL_2(\mathbb C)\times PSL_2(\mathbb C)\to PSL_2(\mathbb C)$$

is surjective. Equivalently, for every $a\in PSL_2(\mathbb C)$, the equation

$$w(x_1,x_2)=a$$

has a solution. This is presented as a challenging problem concerning word maps on semisimple algebraic groups; the supplied source does not state whether it has been resolved.