Extension of the suspension bound to arbitrary trees

Prove that for every tree T whose smaller color class has size k, the Turán number of its suspension satisfies ex(n,Ť) ≤ f(n,k) for all sufficiently large n.

Background

The main theorem is proved for balanced trees, whose two color classes differ in size by at most one. The authors note that the proof would also apply to certain unbalanced trees whenever the structural condition in Lemma 4 holds, including double stars with an appropriate parameter range.

They then formulate a conjecture extending the asymptotic upper bound from balanced trees to every tree, using the size k of the smaller color class.