The virtually abelian image conjecture for unitary braided vector spaces

Let $V$ be a finite-dimensional complex vector space and let $c\in\operatorname{Aut}(V^{\otimes 2})$ be a unitary solution of the Yang–Baxter equation, so that $(V,c)$ is a unitary braided vector space (BVS). For each $n$, let $\rho^c:\mathcal{B}_n\to\operatorname{GL}(V^{\otimes n})$ be the braid-group representation defined by

$$\rho^c(\sigma_i)=I^{\otimes(i-1)}\otimes c\otimes I^{\otimes(n-i-1)}.$$

Virtually abelian image conjecture. For every $n$, the image $\rho^c(\mathcal{B}_n)$ is virtually abelian; moreover, if $c$ has finite order, then $\rho^c(\mathcal{B}_n)$ is finite.

This conjecture seeks a uniform description of braid-group images arising from unitary braided vector spaces. The finite-order case predicts finiteness, while the general case predicts virtual abelianness; the supplied text gives no resolution status.