Coproduct positivity conjecture for the embedded dual canonical basis

Let $\widetilde{{\mathbf U}}^\imath$ be a right coideal subalgebra of $\widetilde{{\mathbf U}}$, with embedding $\iota$, and let $\mathcal{B}^\imath$ and $\mathcal{B}$ be the respective dual canonical bases. Thus $\Delta(\widetilde{{\mathbf U}}^\imath)\subset\widetilde{{\mathbf U}}^\imath\otimes\widetilde{{\mathbf U}}$. Embedded coproduct positivity conjecture. For each $b^\imath\in\mathcal{B}^\imath$, there are coefficients $c^{b^\imath}_{b_1^\imath,b_2}\in\mathbb{N}[v,v^{-1}]$ such that

$$\Delta(\iota(b^\imath))=\sum_{b_1,b_2\in\mathcal{B}}c^{b^\imath}_{b_1^\imath,b_2}b_1^\imath\otimes b_2.$$

This predicts positivity for the coaction of the ambient bialgebra on the embedded dual canonical basis. It is proved for $\widetilde{{\mathbf U}}_v(\mathfrak{sl}_2)$, while the general case remains open.