A conjecture concerning the q-Onsager algebra

The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $W_0, W_1$ and relations called the $q$-Dolan/Grady relations. Recently Baseilhac Kolb obtained a PBW basis for with elements denoted $\lbrace B_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty, \lbrace \alpha_1} \delta} \rbrace_{n=1}^\infty $. In their recent study of current A_q$, Belliard conjecture that there exist W_{-k}\rbrace_{k=0}^\infty, W_{k+1}\rbrace_{k=0}^\infty, G_{k+1} \rbrace_{k=0}^\infty, {\tilde G}_{k+1} \rbrace_{k=0}^\infty$ in satisfy defining A_q$. order to establish this conjecture, it desirable know how second list above are related first above. present paper, we precise relationship give some supporting evidence. This evidence consists computer checks on SageMath due Travis Scrimshaw, proof our homomorphic image universal Askey-Wilson algebra.