Centralizer conjecture for the alternating generators $\tilde G_{k+1}$ of the $q$-Onsager algebra

Let $O_q$ be the $q$-Onsager algebra, and let $\tilde G_{k+1}$, $k\in\mathbb N$, be its alternating generators. For $x\in O_q$, consider the subalgebra generated by $\{\tilde G_{k+1}\}_{k\in\mathbb N}$. Centralizer conjecture for $\tilde G_{k+1}$. The following conditions are equivalent:

$$\text{$x$ commutes with $\tilde G_{k+1}$ for every $k\in\mathbb N$};$$ $$\text{$x$ belongs to the subalgebra generated by }\{\tilde G_{k+1}\}_{k\in\mathbb N}.$$

The claim identifies the centralizer of the family $\{\tilde G_{k+1}\}$ with the subalgebra that it generates. It is posed among directions for future research, and no resolution is supplied in the source.