The no-orbit-shortening conjecture for reduced Hurwitz components

Let $\bar O_k$ be a component at level $k$, and let ${{\mathcal H}}^{{\text{rd}}}_{\bar O_k}$ denote the corresponding reduced Hurwitz-space component. Orbit shortening is the reduction in the lengths of the relevant cusp orbits when passing from the pullback component to its image. No-orbit-shortening conjecture. For $k$ sufficiently large, there is no orbit shortening above

$${{\mathcal H}}^{{\text{rd}}}_{\bar O_k}.$$

The hypothesis is intended to simplify the Riemann–Hurwitz analysis of component genera and cusp data; the source does not state that it has been resolved.