The necklace braid group image-size conjecture for the Ising representation

Let $V$ be the two-dimensional vector space and let

$$R=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&0&1\\0&1&-1&0\\0&1&1&0\\-1&0&0&1\end{pmatrix},$$

with $\rho^R$ the associated representation of the necklace braid group $\mathcal{NB}_n$. For $n\geq 3$, the necklace braid group image-size conjecture asserts

$$\lvert\rho^R(\mathcal{NB}_n)\rvert=n\lvert\rho^R(B\widetilde{A}_n)\rvert=n2^n\lvert\rho^R(\mathcal{B}_n)\rvert.$$

The authors verified this equality for $n\leq 5$, and it predicts that the necklace braid group image can be significantly larger than the ordinary braid group image. Its status beyond the checked cases is left open.