Double-shortcut equivalence-class conjecture for type A hypercube decompositions

Let $W$ be a Coxeter group of type $A$, and fix a Bruhat interval $[u,v]$. Let $z$ and $z'$ be amazing hypercube decompositions of $[u,v]$, and let $DS(z,z')$ be the multiset of $(z,z')$-double shortcuts. Define an equivalence relation by taking the transitive closure of

$$z\sim z'\quad\text{if }DS(z,z')=DS(z',z).$$

Double-shortcut triviality conjecture. The above equivalence relation is trivial, meaning that it has only one equivalence class. This is a conjecture of Ellenberg and McNamara and is stated to imply the Combinatorial Invariance Conjecture for type $A$.