Does instantiation preserve quasidendroids?

Determine whether, for any linear orders α ≤ β and any α-quasidendroid D, the β-instantiation D_β is necessarily a β-quasidendroid; equivalently, ascertain whether D_β can introduce an infinite branch even when D has no infinite branch.

Background

The paper introduces prequasidendroids and quasidendroids and defines the β-instantiation D_β of an α-prequasidendroid D by systematically replacing parameter ordinals via increasing maps. While D_β is always an β-prequasidendroid, the key obstruction to being a β-quasidendroid is the existence of an infinite branch after instantiation.

The authors explicitly note that it is unknown whether being a quasidendroid is preserved under instantiation, because instantiation might, in principle, create an infinite branch even if the original D had none. Establishing preservation (or finding a counterexample) would clarify the stability of the quasidendroid notion under parameter extension and would impact the functorial treatment of proof trees used in β-logic.