Strand-number conjecture for braids of maximal entropy

Let $n\ge 2$, and let $\operatorname{Br}(n)$ be the braid group on $n$ strands. For a braid $\beta$, write $l(\beta)$ for the minimum number of standard generators needed to express it, and call $\beta$ of maximal entropy when

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

Strand-number conjecture. Braids of maximal entropy belong to $\operatorname{Br}(3)$ or $\operatorname{Br}(4)$. The paper presents this as an empirical conjecture obtained from its computational program.