Nonexistence conjecture for surjections between braid groups of higher-genus surfaces

Let $S=S_{g,p}$ be a surface with $p\ge 0$ and $g>1$. For positive integers $m$ and $n$ with $m\neq n$, consider homomorphisms between the braid groups $B_n(S)$ and $B_m(S)$. Nonexistence conjecture. There is no surjective homomorphism

$$F:B_n(S)\to B_m(S).$$

This would extend the classification of surjective homomorphisms in the pure braid-group setting and the known obstruction from braid groups to pure braid groups. The conjecture concerns the remaining classification problem for homomorphisms between braid groups of surfaces of genus greater than one; the supplied context does not indicate whether it has been resolved.