Equality for every order in the suspension Turán bound

Determine whether equality ex(n, ĤT)=f(n,k) holds for every n for balanced trees T whose smaller color class has size k but which do not contain a matching covering one color class.

Background

Theorem 3 gives the upper bound ex(n, ĤT)≤f(n,k) for sufficiently large n and shows that equality holds for infinitely many n. Proposition 6 establishes equality for every n when T contains a matching of size k, equivalently a matching covering the smaller color class.

For other balanced trees, the authors do not know whether equality holds for every n. They explain that resolving this question would require constructing suitable (k−1)-regular, suspension-decomposition-family-free graphs of the relevant order.