Conjecture on great-circle thrackleable trees and augmented caterpillars

A great-circle thrackleable tree is a tree admitting a thrackle drawing on the sphere in which the edges are great-circle arcs. An augmented caterpillar is the graph class defined in the source. Conjecture on great-circle thrackleable trees and augmented caterpillars. The class of great-circle thrackleable trees is properly contained in the class of augmented caterpillars. The conjecture is proposed as a further restriction on the class of trees that can be great-circle thrackleable; the source leaves it open and notes that crossing orientations may be relevant to proving or disproving it.