Nonclassical iota-canonical basis conjecture

Let $(V,\mathbb{B}_V)$ be a based $U_{-q^2}$-module, for example a finite-dimensional irreducible type I representation with Lusztig canonical basis $\mathbb{B}_V$. Let $\mathcal{N}=\mathbb{C}(q)\{1_{\mathcal{N}}\}$ be the one-dimensional nonclassical representation, and let $\Theta^\iota$ be the $2$-tensor quasi $K$-matrix acting on $V\otimes\mathcal{N}$. Nonclassical iota-canonical basis conjecture. The operators satisfy

$$\Theta^\iota=(T_{w_0}^{-1}\otimes\prescript{\iota}{}{T}_{w_0}^{-1})\circ\Delta(\prescript{\iota}{}{T}_{w_0}),$$

where $T_{w_0}$ and $\prescript{\iota}{}{T}_{w_0}$ are the longest quantum Weyl group elements; $\Theta^\iota$ induces an $\iota$bar involution, there is a unique $\iota$canonical basis $\mathbb{B}_{V\otimes\mathcal{N}}^\iota$ that is invariant and unitriangular relative to $\mathbb{B}_V\otimes1_{\mathcal{N}}$, this basis respects isotypic components, and a compatible nonclassical $\iota$canonical basis exists for a modified form of $U_{-q^2}^\iota$. In rank $1$, that basis consists of the nonclassical $\iota$divided powers $\mathsf{x}^{(k)}$. These assertions extend the usual canonical-basis formalism to the nonclassical quantum symmetric-pair setting. The statements are presented as conjectural; the rank-one description and preceding operator properties provide partial evidence, while the general existence and compatibility claims remain open.