Bounded comparison for odd-crossing and crossing numbers

Establish whether there is a universal constant c > 0 such that the odd-crossing number of every graph is at most c times its crossing number.

Background

The odd-crossing number counts pairs of edges that cross an odd number of times and is bounded above by the pair-crossing number, which in turn is bounded above by the ordinary crossing number under the conventions used in the paper.

Although examples show that these parameters can differ, the paper does not know whether their ratio is universally bounded by a constant. The best stated general estimate is quadratic rather than linear.