Negami's joint crossing number conjecture

Let $G_1$ and $G_2$ be graphs embedded on a closed surface $S$, and let the joint crossing number be the minimum number of crossing points between $h(G_1)$ and $G_2$ over all homeomorphisms $h:S\to S$. Negami's conjecture. There exists a universal constant $C$ such that, for any such pair, the joint crossing number is at most

$$C|E(G_1)||E(G_2)|.$$

Negami proved an $O(g|E(G_1)||E(G_2)|)$ bound for graphs embedded on a closed surface of genus $g$. The conjecture would give a genus-independent bound and is equivalent to the existence of short decompositions of any fixed topological shape; it remains open.