Boundary-orbit conjecture for atoms of the compactified outer billiard map

Recall that the sets $\widehat A$ and $\widehat B$ are partitioned into atoms on whose interiors the first-return map of $\widehat\Psi$ is defined. Boundary-orbit conjecture. No point on the boundary of an atom has a well-defined $\widehat\Psi$ orbit. This conjecture concerns the behavior of the compactified outer billiard system at the discontinuity boundaries of its atom partition; the source presents it as being motivated by experimental evidence, without giving a resolution.