Ackerman's linear extremal conjecture for forbidden circle-graph matchings

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A circle graph $F$ is the intersection graph of chords on a circle. A matching in a geometric graph has an intersection graph whose vertices correspond to the matching edges, with adjacency when the corresponding edges cross. Ackerman's linear extremal conjecture for forbidden circle-graph matchings. For any circle graph $F$ on $k$ vertices, there exists a constant $c_k$ such that every $n$-vertex geometric graph that does not contain a matching whose intersection graph is $F$ contains at most $c_kn$ edges. This extends the paper's established linear bound for the relevant three-vertex case and would give a uniform linear extremal bound for every fixed circle graph. The general assertion is presented as a conjecture and no resolution is supplied here.