The finite-image conjecture for unitary finite-order Yang–Baxter solutions

Let $V$ be a finite-dimensional complex vector space, and let $R$ be a unitary solution to the Yang–Baxter equation on $V^{\otimes 2}$. Suppose that $R$ has finite projective order, and let $(\rho_R,V^{\otimes n})$ be the corresponding representations of the braid groups $\mathcal{B}_n$. Finite-image conjecture. The image $\rho_R(\mathcal{B}_n)$ is a finite group projectively for every $n$. If unitarity or finite projective order is omitted, the assertion is false; the conjecture relates finite-order unitary braid representations to property $\mathbf{F}$.