Braid group automorphism conjecture for the universal b9c quantum group

Let $\widetilde{{\mathbf U}}^\imath$ be the universal $\imath$quantum group with generators $B_j$ and $\widetilde{k}_j$, indexed by $j\in\mathbb I$. Let $\tau$ be the diagram involution and $w_\bullet$ the longest element associated with the finite-type part. For $i\in\mathbb I$ satisfying $\tau i=i=w_\bullet i$, and with $e\in\{1,-1\}$, let $\widetilde{y}_{i,j;1,-a_{ij},\overline{p},\overline{t},e}$ and $\widetilde{y}'_{i,j;1,-a_{ij},\overline{p},\overline{t},e}$ be the elements defined by the displayed formulas in the paper.

Braid group automorphism conjecture. For every such $i$, there exist mutually inverse algebra automorphisms $\mathbf T_{i,e}'$ and $\mathbf T_{i,-e}”$ of $\widetilde{{\mathbf U}}^\imath$, determined by

$$\mathbf T_{i,e}'(\widetilde{k}_j)=(-q_i^{1+e}\widetilde{k}_i)^{-a_{ij}}\widetilde{k}_j=\mathbf T_{i,-e}”(\widetilde{k}_j),\qquad \forall j\in\mathbb I,$$

and

$$\mathbf T_{i,e}'(B_j)=\begin{cases}\widetilde{y}_{i,j;1,-a_{ij},\overline{p},\overline{t},e},&j\ne i,\\(-q_i^{1+e}\widetilde{k}_i)^{-1}B_i,&j=i,\end{cases}$$ $$\mathbf T_{i,-e}”(B_j)=\begin{cases}\widetilde{y}'_{i,j;1,-a_{ij},\overline{p},\overline{t},e},&j\ne i,\\(-q_i^{1+e}\widetilde{k}_i)^{-1}B_i,&j=i.\end{cases}$$

These maps should therefore be well-defined on all defining relations of $\widetilde{{\mathbf U}}^\imath$.

The proposed maps are $\imath$-quantum-group analogues of Lusztig's braid group symmetries. The paper notes that even in finite type no braid group action on $\mathbf U^\imath$-modules is available, and direct verification that these maps are algebra automorphisms is difficult; the conjecture's resolution is not supplied here.