Alternative conditions yielding a Crossing Lemma for multigraphs

Identify a natural condition on drawings of multigraphs, distinct from the non-homotopic condition, that guarantees a Crossing Lemma-type lower bound on the number of crossings in terms of the numbers of vertices and edges.

Background

The ordinary Crossing Lemma fails for unrestricted multigraphs because arbitrarily many parallel edges or loops can be drawn without crossings. The paper surveys several sufficient drawing restrictions, including bounded multiplicity, separated drawings, single-crossing drawings, locally starlike drawings, and non-homotopic drawings.

The authors explicitly ask whether another natural drawing condition can yield a comparable Crossing Lemma-type statement, suggesting that the known conditions do not exhaust the relevant possibilities.