Bao–Wang's asymptotic stability conjecture for trivial submodules

Let $\lambda\in X^+$ satisfy $\overline{\lambda}=\overline{0}$, and let $w_0\in V(\lambda)$ be the vector, unique up to nonzero scalar multiple, whose $\mathbf{U}^\imath$-submodule is isomorphic to the trivial module $V(0)$. Write $v_\lambda$ for the highest-weight vector and let $\equiv_\infty$ denote the asymptotic equivalence used for the \imathcanonical-basis construction. Asymptotic stability conjecture. There exist parameters ${\boldsymbol \varsigma}$ and ${\boldsymbol \kappa}$ such that there is a scalar $c\in\mathbb{Q}(q)^\times$ with

$$cw_0\equiv_\infty v_\lambda.$$

This is a further stability assertion relating the trivial submodule generator to the highest-weight vector. The source gives no resolution status, and the notation $\equiv_\infty$ is defined elsewhere in the paper.