Bijection conjecture for the q-Onsager algebra presentation

Let ${\cal O}_q$ be the $q$-Onsager algebra and let $\tilde{{\cal A}}_q^{\{\delta\}}$ be the algebra generated by ${\cal W}_0$ and ${\cal W}_1$. Define

$$\Psi:{\cal O}_q\longrightarrow\tilde{{\cal A}}_q^{\{\delta\}},\qquad \Psi({\textsf A})={\cal W}_0,\quad \Psi({\textsf A}^*)={\cal W}_1.$$

Bijection conjecture. The map $\Psi$ is a bijection.

The map is already known to be a surjective homomorphism, so the conjecture asserts injectivity and hence identifies the two algebras. The supplied text gives no resolution.