A conjecture concerning the $q$-Onsager algebra

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