An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra

Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms $T_0$, $T_1$ of the $q$-Onsager algebra $\mathcal O_q$, that are roughly analogous to the Lusztig automorphisms of $U_q(\widehat{\mathfrak{sl}}_2)$. We use $T_0, T_1$ and a certain antiautomorphism of $\mathcal O_q$ to obtain an action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on $\mathcal O_q$ as a group of (auto/antiauto)-morphisms. The action forms a pattern much more symmetric than expected. We show that a similar phenomenon occurs for the associated current algebra $\mathcal A_q$. We give some conjectures and problems concerning $\mathcal O_q$ and $\mathcal A_q$.