Diagonal conjugation conjecture for match Yang–Baxter representations

Let $N\in\mathbb{N}$, let $(1,R)\in YB^1(\mathsf{Match}^N)$, and let $X$ be a diagonal matrix in $\mathsf{Mat}^N(2,2)$. Define

$$S=XRX^{-1}.$$

For each $n$, let $\rho_n^R$ and $\rho_n^S$ denote the associated braid-group representations. Diagonal conjugation conjecture. There exists a sequence of diagonal matrices $A_n\in\mathsf{Mat}^N(n,n)$, independent of $R$ and depending only on $X$, such that

$$A_n\rho_n^R=\rho_n^S A_n$$

for all $n$. Consequently, the assignment $R\mapsto XRX^{-1}$ lifts to an inner autoequivalence of $\mathsf{MoonFun}^1(\mathsf{B},\mathsf{Match}^N)$ and, in particular, to an $\infty$-equivalence. This conjecture formalises the suspected $X$-symmetry of match Yang–Baxter representations: low-rank testing suggests the claimed equivalence, but a general construction of the matrices $A_n$ is not established.