Finite-order injectivity conjecture for inverse semigroups

Let $S$ be an inverse semigroup, let $\alpha\in\operatorname{Aut}(S)$ have finite order, and suppose that the map

$$x\mapsto x^{-1}\cdot x\alpha$$

is injective. Here $E(S)$ denotes the set of idempotents of $S$, and $\operatorname{Fix}(\alpha)$ denotes the set of elements fixed by $\alpha$.

Finite-order injectivity conjecture. Then

$$\operatorname{Fix}(\alpha)=E(S).$$

The general injectivity problem asks whether this equality follows without the finite-order hypothesis; the paper reports computational evidence affirming it when $\alpha$ has finite order.