GGS conjecture for explicit quantum R-matrices of $\mathfrak{sl}_n$

Let $(\Gamma_1,\Gamma_2,\tau)$ be an admissible triple and let $r^0\in\mathfrak h\otimes\mathfrak h$ satisfy the equations referenced in the source. Define $\tilde r^0=r^0-P^0/2$, and construct $R\in\operatorname{Mat}_n(\mathbb C)\otimes\operatorname{Mat}_n(\mathbb C)$ by

$$R=q^{\tilde r^0}(R_s+\hat q\,\tilde a)q^{\tilde r^0},$$

where $\tilde a$, $R_s$, and $\hat q=q-q^{-1}$ are defined as above. Here $P$ denotes the permutation operator, and $R_{12},R_{13},R_{23}$ are the corresponding tensor placements. GGS conjecture. The matrix $R$ satisfies the quantum Yang–Baxter equation

$$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12},$$

and $PR$ satisfies the Hecke relation

$$(PR-q)(PR+q^{-1})=0.$$

This conjecture gives an explicit quantization of the Belavin–Drinfeld solutions of the classical Yang–Baxter equation. The supplied source is a verification for $\mathfrak{sl}_n$ with $n\leq 12$, so the general claim beyond the verified range is not established by this paper.