Damiani's PBW-basis conjecture for the $q$-Onsager algebra

Let $O_q$ be the $q$-Onsager algebra, with alternating generators $\{W_{-i}\}_{i\in\mathbb N}$, $\{\tilde G_{j+1}\}_{j\in\mathbb N}$, and $\{W_{k+1}\}_{k\in\mathbb N}$. A linear order $<$ on these generators is said to satisfy one of the six prescribed orderings:

$$W_{-i}<\tilde G_{j+1}<W_{k+1},\quad W_{k+1}<\tilde G_{j+1}<W_{-i},\quad W_{k+1}<W_{-i}<\tilde G_{j+1},$$ $$W_{-i}<W_{k+1}<\tilde G_{j+1},\quad \tilde G_{j+1}<W_{k+1}<W_{-i},\quad \tilde G_{j+1}<W_{-i}<W_{k+1},$$

for all $i,j,k\in\mathbb N$. Damiani's PBW-basis conjecture. For any such linear order, the ordered monomials in these alternating generators form a PBW basis for $O_q$. This would provide a PBW-type description of the $q$-Onsager algebra compatible with all six block orderings; the source presents it as a variation on an earlier conjecture and does not give evidence of a resolution.