The hypercube-decomposition conjecture for Kazhdan–Lusztig polynomials

Let $W$ be a Weyl group of type $A$, and let $u,v\in W$ satisfy $u\leq v$. A subinterval $[u,z]$ is a hypercube decomposition of $[u,v]$ when it has the diamond-completeness and hypercube-cluster properties described in the source, and let $\widetilde{P}_{u,v,z}(q)$ be the polynomial produced by the associated procedure.

Hypercube-decomposition conjecture. For any hypercube decomposition $[u,z]$ of $[u,v]$,

$$P_{u,v}(q)=\widetilde{P}_{u,v,z}(q).$$

The conjecture is proposed for type $A$ Weyl groups and concerns an explicit algorithm for computing Kazhdan–Lusztig polynomials; the supplied text gives no resolution.