The measure-zero conjecture for minimal restricted interval exchange transformations

Let $\pi$ be a permutation, let $\langle\,,\rangle_\pi$ be the associated pairing, and let $\mathscr R$ be a restriction space that is rich with respect to $\langle\,,\rangle_\pi$. Write $X_{\pi,\mathscr R}$ for the space of interval exchange transformations satisfying the restrictions in $\mathscr R$, and let $M_{\pi,\mathscr R}$ be its subset of minimal transformations.

Measure-zero conjecture. The subset $M_{\pi,\mathscr R}$ has zero Lebesgue measure in $X_{\pi,\mathscr R}$.

This conjecture strengthens the preceding expectation that minimal transformations satisfying a rich restriction space are not stably minimal. It is motivated by examples in which the minimal locus has codimension one, or has codimension between zero and one; the general measure-zero statement remains open.