Little's conjecture on dual Edelman–Greene equivalence

Let $R(\infty)$ be the set of all reduced words of permutations. Two reduced words are dual $EG$-equivalent when their recording tableaux under Edelman–Greene insertion agree, that is, $Q({\mathbf{i}})=Q({\mathbf{i}}')$. Little's conjecture. Two reduced words are dual $EG$-equivalent if and only if they are connected by forward and backward Little moves. This conjecture characterizes dual Edelman–Greene equivalence through the combinatorics of Little moves.