The maximal braid-edge density conjecture for layered permutations

Let $d53e$ be the symmetric group on $n$ letters, and let $w_0^{(k,i)}$ denote the family of permutations specified in the paper. For $d70e i d53e$, let $G(d70e)$ be the graph of reduced words, with vertices corresponding to reduced words and braid edges corresponding to braid moves. The maximal braid-edge density conjecture. For all $d70e i d53e$, if $d70e=w_0^{(k,i)}$ for some $k,i i ad53c$, then $G(d70e)$ has the highest proportion of braid edges to vertices among all graphs of $d4a1(d70e)$ in $d53e$. The claim is a proposed second route toward the paper's main theorem and is not resolved in the supplied text.