Minimality conjecture for interval exchange transformations with flips

Let $\mathbb{S}^1$ be the circle, let $n$ be any positive integer, and let $F\in\operatorname{CET}_n$ be an interval exchange transformation with flips whose flip parameter is $\tau$. Minimality conjecture. If $F$ is minimal on $\mathbb{S}^1$, then

$$\tau=\frac{1}{2}.$$

The source presents this as a general question about minimality for the modified Rauzy graphs and does not give a resolution.