The lower-generator uniqueness conjecture for the q-Onsager algebra

Let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, and let $\mathcal J$ be its two-sided ideal generated by $A$ and $B$. Lower-generator uniqueness conjecture. There exists a unique sequence $\{\tilde g_{k+1}\}_{k\in\mathbb N}$ of elements in $\mathcal J$ satisfying the relations in the presentation conjecture for $\mathcal A_q$, with $\mathcal W_0=A$ and $\mathcal W_1=B$.

The claim asserts existence and uniqueness of lower generators inside the augmentation ideal of the q-Onsager algebra. No resolution is supplied.