Conjecture on the size of conjugacy classes of canonical idempotents in free inverse semigroups

Let $X$ be an alphabet, let $\e\in\mathcal{FI}(X)$ be a canonical idempotent, and let $|e|$ denote its word length. The conjugacy class of $e$ is the equivalence class $[e]_{\mathrel{\sim_i}}$ under inverse-semigroup conjugacy.

Conjugacy-class cardinality conjecture. The conjugacy class of $e$ has

$$\frac{|e|}{2}+1$$

elements.

The preceding result establishes only that this conjugacy class is finite; the exact cardinality asserted here remains to be proved.