Positivity conjecture for dual canonical bases in type AIII

Consider the $\imath$quantum group of type ${\rm AIII}_{2r}$ and its dual canonical basis. Type-${\rm AIII}_{2r}$ positivity conjecture. The positivity results for the dual canonical basis still hold in type ${\rm AIII}_{2r}$; in particular, the structural constants for multiplication belong to $\mathbb{N}[v,v^{-1}]$. This extends the positivity expectations beyond the cases treated by the existing construction; the supplied text does not state a resolution, so the conjecture remains open.