Existence of continuous non-trivial embeddings of minimal interval exchange transformations

Let $d\geq 3$. A minimal $d$-IET is an interval exchange transformation $(I,f)$ with $d$ exchanged subintervals, whose dynamics is minimal; a $d$-PWI is a planar piecewise isometry with $d$ atoms. An embedding is non-trivial when it is not one of the trivial embeddings considered in the paper.

Existence conjecture. For any $d\geq 3$, there is a minimal $d$-IET $(I,f)$ which admits a continuous, non-trivial embedding into a $d$-PWI.

The conjecture asks whether continuous non-trivial embeddings exist in every dimension at least three. The paper establishes obstructions in dimension two and strong restrictions for three-atom PWIs, while examples suggest that four-atom PWIs can admit many such embeddings.