The upper bound conjecture for ergodic invariant measures of FIETs

Let a flipped interval exchange transformation (FIET) be an interval exchange transformation with flips, and let $n$ be the number of exchanged intervals. An invariant ergodic measure is an invariant probability measure that is ergodic under the FIET.

Upper bound conjecture. A FIET can have at most

$$\left\lfloor \frac{n}{2} \right\rfloor -1$$

distinct invariant ergodic measures.

This hypothesis proposes a sharp upper bound on the number of distinct invariant ergodic measures for FIETs. The supplied text does not establish whether the bound is known in general or remains open.