The hare–tortoise fertility gap conjecture

Let $w$ be a word of length $m$, and let $\langle w\rangle_{\operatorname{\mathsf{hare}}}$ and $\langle w\rangle_{\operatorname{\mathsf{tortoise}}}$ denote its hare- and tortoise-fertilities. The hare–tortoise fertility gap conjecture.

$$\langle w\rangle_{\operatorname{\mathsf{hare}}}-\langle w\rangle_{\operatorname{\mathsf{tortoise}}}\leq\frac{m-5}{2}.$$

The preceding theorem gives examples of length $2k+5$ attaining a gap of $k$, and the conjecture asserts that these examples are minimal with respect to length.