The relative braid group fixed-subgroup conjecture

Let $f_\Gamma:\Sigma\to\underline \Sigma=\Sigma/\Gamma$ be the quotient map in the assumptions of Proposition. Write $Br_\Sigma^\Gamma$ for the subgroup of $Br_\Sigma$ fixed by $\Gamma$, and $Br_\Sigma^{f_\Gamma}$ for the relative braid group. Relative fixed-subgroup conjecture. In the assumptions of Proposition,

$$Br_\Sigma^{f_\Gamma}=Br_\Sigma^\Gamma.$$

The inclusion $Br_\Sigma^{f_\Gamma}\subseteq Br_\Sigma^\Gamma$ is immediate from the proposition; the conjecture asserts the reverse inclusion.