Exact Turán number for non-nested matchings

Determine whether, for every pair of integers n,k with n≥2k, the Turán numbers for non-nested and strongly non-nested k-matchings both equal (k−1)n; that is, prove whether ex_<(n,¬Nest-𝓜_k)=ex_<(n,¬Nest*-𝓜_k)=(k−1)n.

Background

The paper establishes the bounds (k−1)n≤ex_<(n,¬Nest-𝓜k)≤ex<(n,¬Nest*-𝓜_k)≤(k−1)n+\binom{k−1}{2} for n≥2k. A strongly non-nested matching is a disjoint union of crossing matchings in which two edges cross if and only if they belong to the same crossing matching, so every strongly non-nested matching is non-nested.

The conjecture asserts that the lower bound is exact for both extremal problems, thereby determining the Turán number for non-nested matchings and showing that restricting to strongly non-nested matchings does not change the extremal value. The conjectured formula is also used later in the paper to derive an improved upper bound for the corresponding ordered Ramsey number.