Higher spin Serre relations

Let $A_m^n$ be a unital ${\mathbb C}(q)$-algebra, and let $\{\mathsf X_r^{(i)}\}_{1\leq r\leq m-1,\,0\leq i\leq n}\subset A_m^n$ satisfy $\mathsf X_r^{(0)}=1$ together with the relations defining the elements $\mathsf X_r^{(i)}$ and the spin Serre relations. Higher spin Serre conjecture. For $1\leq a,b,c\leq n$, there are $\xi\in{\mathbb C}(q)$ and $1\leq a',b',c'\leq n$ such that

$$\mathsf X_i^{(a)}\mathsf X_{i\pm1}^{(b)}\mathsf X_i^{(c)}=\xi\,\mathsf X_{i\pm1}^{(a')}\mathsf X_i^{(b')}\mathsf X_{i\pm1}^{(c')}+\mathrm{LOT}_{a,b,c},$$

where $\mathrm{LOT}_{a,b,c}$ is a linear combination of terms $\mathsf X_i^{(k)}\mathsf X_{i\pm1}^{(\ell)}$ and $\mathsf X_{i\pm1}^{(\ell)}\mathsf X_i^{(k)}$. This is intended to provide more-general versions of the stated spin Serre proposition and is used in the proposed characterization of spin link polynomials.