The monotonicity conjecture for exceptional-word ratios

For each positive integer $m$, let $\mathcal E_m$ be the set of exceptional normalized words of length $m$, and let $\mathcal{NW}_m$ be the set of normalized words of length $m$. Here, a word is exceptional when its hare-fertility exceeds its tortoise-fertility. The monotonicity conjecture for exceptional-word ratios. The numbers

$$\frac{|\mathcal E_m|}{|\mathcal{NW}_m|}$$

are increasing in $m$. The paper reports computed values for $m=7,8,9$ and observes that, if the conjecture holds, the limiting ratio exists; its value remains to be determined.