Bao–Wang's stability conjecture for imathcanonical bases

Let $\mathbf{U}^\imath$ be the \imathquantum group with parameters ${\boldsymbol \varsigma}$ and ${\boldsymbol \kappa}$. For each $\lambda,\mu,\nu\in X^+$, let $V^\imath(\lambda,\mu;\nu)$ be the corresponding based $\mathbf{U}^\imath$-module, and let

$$\dot{\pi}^\imath_{\lambda,\mu;\nu}:\dot{\mathbf{U}}^\imath\longrightarrow V^\imath(\lambda,\mu;\nu)$$

be the homomorphism defined by $x\mapsto xv^\imath_{\lambda,\mu;\nu}$. Stability conjecture. There exist parameters ${\boldsymbol \varsigma}$ and ${\boldsymbol \kappa}$ such that the homomorphisms $\dot{\pi}^\imath_{\lambda,\mu;\nu}$ are based for all $\lambda,\mu,\nu\in X^+$. This asserts the desired stability of the \imathcanonical bases under the maps from the modified \imathquantum group; the source supplies no resolution status.