Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\to S_n$ be the canonical epimorphism.
Does every homomorphism $f:\Sigma_g\to S_n$ lift to $B_n$? That is, is there a homomorphism $\bar f:\Sigma_g\to B_n$ such that $f=p\circ\bar f$?
This is known for $n=3$, the proof is by ad hoc elementary computations (Hector, Meigniez, Matsumoto, "Ends of leaves of Lie foliations", J. Math. Soc. Japan 57 (2005), no. 3, 753--779.) Is it true for every $n$? The question is crucial for the construction of some Lie foliations.