The linear disjoint-matching conjecture for angularly monotone topological graphs

A simple angularly monotone topological graph is a topological graph drawn on a cylinder whose base is horizontal, in which every vertical line intersects every edge at most once; it is complete when every pair of vertices is joined by an edge. A disjoint matching is a set of pairwise vertex-disjoint edges. For a graph on $n$ vertices, let the matching size be the number of edges in the matching.

Angularly monotone matching conjecture. Every simple angularly monotone complete topological graph on $n$ vertices contains a disjoint matching of size

$$\Omega(n).$$

This would be the angularly monotone analogue of the immediate linear bound for complete $x$-monotone topological graphs, where a matching of size $\left\lfloor n/2\right\rfloor$ exists. Together with the paper's reduction from general simple complete topological graphs to cylindrical angularly monotone drawings, the conjecture could improve the known $\Omega(n^{1/3})$ lower bound for disjoint matchings in complete simple topological graphs to a stronger bound.