Centralizer rigidity conjecture for powers of interval exchange transformations

Let $T$ be an interval exchange transformation satisfying the hypotheses of Theorem 3.4. Write $C(T)$ for its centralizer.

Centralizer rigidity conjecture. For every integer $n\geq 2$,

$$C(T^n)=C(T)=\langle T\rangle.$$

The conjecture would give a negative answer to the questions of whether $T^2$ can have a square root different from $T$ and whether its centralizer can contain torsion. These questions remain unanswered by the results of the paper.