The GGS conjecture for the GGS R-matrix

Let $R_{\text{GGS}}$ be the matrix defined by

$$R_{\text{GGS}}=\sum_{i,j,k,l}P_{i,j,k,l}e_{ij}\otimes e_{kl},$$

where the polynomials $P_{i,j,k,l}$ are uniquely determined by

$$\sum_{i,j,k,l}P_{i,j,k,l}e_{ij}\otimes e_{kl}\equiv 1+2\hbar r+2\hbar^2r^2\pmod{\hbar^3}.$$

Here $q=e^{\hbar}$, and the quantum Yang–Baxter equation is

$$R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12},$$

while the Hecke relation is

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

with $P$ the permutation operator. The GGS conjecture. The matrix $R_{\text{GGS}}$ satisfies the quantum Yang–Baxter equation and the Hecke relation.

These conditions assert that the quantization determined to second order by the classical matrix $r$ extends to a quantum $R$-matrix of Hecke type. The source provides the conjecture but no resolution status, so it is recorded as open.