Alternation conjecture for braids of maximal entropy

Let $n\ge 2$ and let $\beta\in\operatorname{Br}(n)$. If

$$\beta=\sigma_{i_1}^{\epsilon_1}\cdots\sigma_{i_k}^{\epsilon_k}$$

with $i_{k+1}=i_k\pm1$, $|\epsilon_k|=1$, and $\epsilon_{k+1}=-\epsilon_k$, call $\beta$ alternated with respect to the standard generators. A braid is of maximal entropy when

$$h(\beta)=\max\{h(\beta')\mid l(\beta')\le l(\beta)\}.$$

Alternation conjecture. Braids with maximal entropy are alternated. The conjecture concerns the observed structure of entropy-maximizing braids among those considered by word length.