Amazing R-element conjecture for double-shortcut equivalence classes

Let $W$ be a Coxeter group of type $A$, and fix a Bruhat interval $[u,v]$. Consider the equivalence relation on the amazing hypercube decompositions of $[u,v]$ generated by

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

where $DS(z,z')$ is the multiset of $(z,z')$-double shortcuts. Amazing $R$-element conjecture. Every equivalence class of this relation contains an amazing $R$-element. This is a weakening of the conjecture that the relation is trivial; together, these double-shortcut conjectures are intended to imply the Combinatorial Invariance Conjecture for type $A$.