The type-BC indifference graph equivalence conjecture for Kazhdan–Lusztig basis elements

Let $v,w\in\mathfrak{B}_n$ avoid the patterns $3412$ and $4231$. Write $\Gamma(v)$ and $\Gamma(w)$ for their type-$\mathsf{BC}$ indifference graphs, and let $\widetilde{C}_{v}^{\mathsf{BC}}(q),\widetilde{C}_{w}^{\mathsf{BC}}(q)$ be the corresponding Kazhdan–Lusztig basis elements. For elements of $H_n^{\mathfrak{B}}(q)$, define $D_1\approx_qD_2$ when every trace $\theta_q\in\mathcal{T}(H_n^{\mathfrak{B}}(q))$ has $\theta_q(D_1)=\theta_q(D_2)$. The type-BC indifference graph conjecture. If $\Gamma(v)\cong\Gamma(w)$, then

$$\widetilde{C}_{v}^{\mathsf{BC}}(q)\approx_q\widetilde{C}_{w}^{\mathsf{BC}}(q).$$

The corresponding implication at $q=1$ is known only in the reverse direction, so this strengthening remains open even for the indicated pattern-avoiding elements.