The bounded-boxicity conjecture for large-edgewidth graphs on a fixed surface

Let $G$ be a graph embedded in a fixed surface $\Sigma$. The edgewidth of $G$ is the length, measured by the number of edges, of a shortest noncontractible cycle in $G$.

Large-edgewidth boxicity conjecture. There exists an integer $e_\Sigma$ such that every graph $G$ embeddable on $\Sigma$ with edgewidth at least $e_\Sigma$ has boxicity at most three.

Kawarabayashi and Mohar proved that graphs embeddable in a fixed surface with sufficiently large edgewidth are acyclically 7-colorable, which yields boxicity at most 42. The conjecture proposes the stronger bound three.