Suspension Turán bound for arbitrary trees

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

Background

The main theorem proves the bound ex(n, ĤT)≤f(n,k) for balanced trees, where the two color classes differ in size by at most one. The authors observe that the proof would also apply to certain unbalanced trees whenever the structural condition in Lemma 4 holds; they give double stars as an example.

They then conjecture that the same asymptotic upper bound should hold for every tree, using the size k of its smaller bipartition class.