Equality for every order in the non-perfect-matching cases

Determine whether the equality ex(n,ĤT)=f(n,k) holds for every n for balanced trees T whose relevant color class is not covered by a matching, where ĤT is the suspension of T and f(n,k)=max{n₀n₁+⌊(k−1)n₀²/2⌋: n₀+n₁=n}.

Background

Theorem 3 establishes the upper bound ex(n,ĤT) ≤ f(n,k) for sufficiently large n when T is a balanced tree of order 2k or 2k+1 and the Erdős–Sós Conjecture holds for all subtrees of T. The paper constructs extremal examples attaining equality for infinitely many n, and Proposition 6 proves equality for every n when T contains a matching of size k. For the remaining trees, the authors explicitly leave unresolved whether equality holds for every n.