Turán number forbidding nested and crossing matchings

Determine whether, for every n≥2k, the Turán number ex_<(n,{Nest-𝓜_k,Cross-𝓜_k}) equals ex_<(n,¬Sep-𝓜_k), whose exact value is given by Theorem~\ref{thm:nonsep}.

Background

The paper determines the exact extremal number when nested and separated k-matchings are forbidden, and separately determines ex_<(n,¬Sep-𝓜k), the extremal number when non-separated k-matchings are forbidden. Since a graph with no non-separated k-matching has neither a nested nor a crossing k-matching, ex<(n,¬Sep-𝓜_k) provides a lower bound for the problem forbidding nested and crossing matchings.

The authors state that the nested-and-crossing problem appears more difficult and explicitly leave open whether this lower bound is tight.