Discontinuity characterization at the exceptional rational set

Determine whether every point of the exceptional set of rational numbers $IS^{-}\setminus\{1\}$ is a discontinuity of Moroz’s map from modified Engel expansions to Pierce expansions; equivalently, establish whether the map’s discontinuity set is the entire set $\mathbb{Q}\cap(0,1]$ with the endpoint $1$ removed.

Background

The paper studies the composition of Moroz’s maps that sends modified Engel expansion data to Pierce expansion data. For the specific pair of modified Engel and Pierce expansions considered here, the paper proves that the map is left-continuous everywhere, continuous at every irrational point and at $1$, and has jump discontinuities at every point of the countable dense set of right endpoints of modified Engel fundamental intervals other than $1.

The authors identify this known discontinuity set as a subset of $\mathbb{Q}\cap(0,1]$ and show that every discontinuity is a jump. They leave unresolved whether all remaining points of the exceptional set $IS^{-}\setminus\{1\}$—equivalently, all rational points in $(0,1)$ under the identification made in Proposition 2.4—are discontinuities.