Equality for all orders in the suspension Turán bound

Determine whether equality ex(n, ̂T) = f(n,k) holds for every n for every balanced tree T whose smaller color class has size k but does not contain a matching covering that color class.

Background

Theorem 3 gives the upper bound ex(n, ̂T) ≤ f(n,k) for sufficiently large n, under the Erdős–Sós conjecture for the subtrees of T, while the constructions in Section 2 establish equality for infinitely many n. Equality for every n is proved when T contains a matching covering all vertices in one color class. The authors explicitly leave unresolved whether the equality persists for all n for the remaining balanced trees.