Conjectured exact density of one-crossing odd drawings

Determine whether the maximum number of edges in a simple graph admitting a drawing in which every edge is crossed by at most one other edge an odd number of times equals 4n − 8.

Background

The quantity m^{odd}_k(n) is the maximum number of edges in a simple graph with a drawing in which each edge is crossed by at most k other edges an odd number of times. The paper records the known upper bound m^{odd}_1(n) ≤ 5n − 9.

Karl and Toth conjecture that this upper bound is not tight and that the exact value should instead be 4n − 8. Because the paper presents this as a conjecture rather than a proved result, it qualifies as an explicitly stated unresolved problem.