Turán bound for suspensions of arbitrary trees

Prove that for every tree T whose smaller color class has size k, the inequality ex(n,ĤT) ≤ f(n,k) holds for all sufficiently large n, where ĤT is the suspension of T and f(n,k)=max{n₀n₁+⌊(k−1)n₀²/2⌋: n₀+n₁=n}.

Background

The main theorem is proved for balanced trees, meaning that the two color classes differ in size by at most one. The authors observe that the proof would extend to an unbalanced tree if the structural decomposition supplied by their Lemma 4 were valid for that tree. This motivates a conjecture covering every tree, with k defined as the size of the smaller color class.