Faithfulness conjecture for the lattice representation of the Hopf algebra

Let ${\mathcal U}$ be the Hopf algebra generated by $\\{\mathsf{K}_i,\mathsf{L}_i,\mathsf{E}_i,\mathsf{F}_i\\}_{1\le i<\mathsf{N}}$ with the defining relations in the paper, and let $\kappa\colon{\mathcal U}\to\mathsf{A}^{\otimes\mathsf{N}^2}$ be the algebra homomorphism defined on generators by

$$\mathsf{E}_i\mapsto\sum_{j=1}^{\mathsf{N}}\mathsf{e}_{i,j},\quad \mathsf{F}_i\mapsto\sum_{j=1}^{\mathsf{N}}\mathsf{f}_{i,j},\quad \mathsf{K}_i\mapsto\mathsf{e}_{i,1}\mathsf{f}_{i,1},\quad \mathsf{L}_i\mapsto\mathsf{f}_{i,\mathsf{N}}\mathsf{e}_{i,\mathsf{N}}.$$

Faithfulness conjecture. The homomorphism $\kappa$ is faithful, that is, injective.

This would identify the abstract Hopf algebra with its realization inside the tensor power of the lattice algebra. The source provides no proof or resolution of this claim.