General braid group action conjecture for acyclic quivers

Let $(Q,\tau)$ be an acyclic $\imath$quiver, let $W_{\tau}$ be the restricted Weyl group, and let ${\rm Br}(W_{\tau})$ be the braid group generated by elements $t_i$ for $i\in \mathbb I_{\tau}$. Let $\widetilde{\mathbf U}^{\imath}$ be the corresponding universal $\imath$quantum group, and let $\operatorname{T}_i$ be the automorphism induced by the commutative diagram defining it. General braid group action conjecture. The diagram induces an automorphism $\operatorname{T}_i\in\operatorname{Aut}(\widetilde{\mathbf U}^{\imath})$, and there exists a homomorphism

$${\rm Br}(W_{\tau})\longrightarrow\operatorname{Aut}(\widetilde{\mathbf U}^{\imath}),\qquad t_i\longmapsto\operatorname{T}_i,$$

for all $i\in\mathbb I_{\tau}$. This extends the finite-type braid group action to acyclic $\imath$quivers; the paper explains that validity in general is expected assuming the injectivity conjecture for the Hall algebra realization, while the theorem is established for Dynkin $\imath$quivers.