The Burau representation's surjectivity conjecture

Let $B_n$ be the braid group, let $\beta:B_n\to\Gamma$ be the unreduced Burau representation, and let $\beta':B_n\to\Gamma'$ be its reduced representation, where $\Gamma$ is the unitary group associated with Squier's Hermitian form and $\Gamma'$ is its restriction to the orthogonal complement of the fixed vector. For $n\geqslant 5$, the Burau surjectivity conjecture. There are equalities

$$\beta(B_n)=\Gamma$$

and

$$\beta'(B_n)=\Gamma'.$$

The preceding theorem establishes density of both images in the respective groups for the $s$-adic topology, but whether either image equals the full unitary group remains open.