PBW basis conjecture for the q-Onsager algebra

Let ${\cal O}_q$ be the $q$-Onsager algebra, and let ${\cal W}_0,{\fal W}_1,{\fal G}_{k+1}$ and ${\cal W}_{-k}$ be the generators used for the algebra $\tilde{{\cal A}}_q^{\{\delta\}}$. For nonnegative integer exponents and indices ordered by

$$k_1<\cdots<k_N,\qquad p_1<\cdots<p_P,\qquad l_1<\cdots<l_M,$$

consider the ordered monomials

$${\cal W}_{-k_1}^{\alpha_1}\cdots {\cal W}_{-k_N}^{\alpha_N}{\cal G}_{p_1+1}^{\beta_1}\cdots {\cal G}_{p_P+1}^{\beta_P}{\cal W}_{l_M+1}^{\gamma_M}\cdots {\cal W}_{l_1+1}^{\gamma_1}.$$

PBW basis conjecture. These vectors induce a Poincare–Birkhoff–Witt-type basis of $\tilde{{\cal A}}_q^{\{\delta\}}$, called the $WG$-basis.

The conjecture is supported by explicit calculations in low degrees. The subsequent comparison with the zig-zag basis suggests that this basis should also yield a basis for the $q$-Onsager algebra, but the supplied text does not establish that consequence.