Inversion conjecture for uniquely 2-divisible completely regular semigroups

Let $S$ be a uniquely $2$-divisible completely regular semigroup. Let $\alpha\in\operatorname{Aut}(S)$ satisfy

$$\alpha^2=1$$

and suppose that $\operatorname{Fix}(\alpha)=E(S)$, where $E(S)$ is the set of idempotents of $S$ and $\operatorname{Fix}(\alpha)$ is the set of elements fixed by $\alpha$.

Inversion conjecture. Then

$$x\alpha=x^{-1}$$

for every $x\in S$.

This proposes an analogue for completely regular semigroups of results proved earlier in the paper for inverse semigroups. The statement is posed as a natural extension, and no resolution is given in the supplied text.