The nonnegative excess conjecture for braid edges

Let $d53e$ be the symmetric group on $n$ letters. For $d70e i d53e$, let $G(d70e)$ be the graph of reduced words of $d70e$, let $d_\mathcal{B}(v)$ denote the braid degree of a vertex $v$, and define

$$A(\sigma)=|\mathcal{R}(\sigma)|-\left(\sum_{v\in G(\sigma)}d_\mathcal{B}(v)\right).$$

The nonnegative excess conjecture. For any $d70e i d53e$,

$$A(\sigma)\geq 0.$$

The text presents this as one of two possible results that would prove the main theorem for all permutations. It is established only for the permutations covered by the preceding lemma, so the general statement remains open.