The $H^1$–$H^r$ equivalence conjecture for low-dimensional permutations

Let $S_{n+1}$ be the symmetric group, let ${\mathbf W}_n$ be the set of words, and let $\operatorname{inv}(\sigma)$ denote the inversion number of $\sigma$. For a nonidentity permutation $\sigma$, set $\dim(\sigma)=\operatorname{inv}(\sigma)-1$. The relations $w\dashv\sigma\,[H^1]$ and $w\dashv\sigma\,[H^r]$ describe the corresponding incidence relations for the $H^1$ and $H^r$ spaces. $H^1$–$H^r$ equivalence conjecture. For $\dim(\sigma)<n$ and all $w\in{\mathbf W}_n$, we have $w\dashv\sigma\,[H^1]$ (that is, $w\preceq(\sigma)$) if and only if $w\dashv\sigma\,[H^r]$ for every $r>2$. The statement is known for $n=2$; the source presents the extension to the stated range as an open question.