Realization of invariant subcontinua as quotients

Determine whether every Φ-invariant subcontinuum of the universal hyperbolic extension Z is equal to L(g) for some representative g in the isotopy class of the pseudo-Anosov map f.

Background

The universal hyperbolic extension contains quotient sets L(g) for all representatives in the isotopy class, and the appendix describes these as invariant subsets of the ambient space. It remains unresolved whether the collection of such quotient sets exhausts all Φ-invariant subcontinua.