On surjectivity of word maps on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2$

Let $w=[[x^k,y^l],[x^m,y^n]]$ be a non-trivial double commutator word. We show that $w$ is surjective on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(K)$, where $K$ is an algebraically closed field of characteristic $0$.