The exceptional Hurwitz structure set conjecture for primitive NET maps

A NET map is a nearly Euclidean Thurston map. A NET map $f$ is primitive if it is not imprimitive, where imprimitive means that there exist NET maps $f_1$ and $f_2$ such that $f_1$ is Euclidean, $f=f_1\circ f_2$, and the postcritical sets of $f$, $f_1$, and $f_2$ are equal. The Hurwitz structure set is the associated equivalence-class datum for a NET map; MC2 and MC4 denote the two specified types of Hurwitz structure sets, and the five exceptional sets are the five representatives listed immediately before the conjecture.

Exceptional Hurwitz structure set conjecture. The Hurwitz structure set of every primitive NET map with constant pullback map is either an MC2 or MC4 Hurwitz structure set, or it is equivalent to one of the five exceptional Hurwitz structure sets listed above.

The claim seeks a complete classification of primitive NET maps with constant pullback maps. The preceding discussion records five exceptional equivalence classes, while the supplied text does not establish that the list is exhaustive.