Quadratic bound between surface crossing numbers

Let $G$ be a graph, and for a surface $[200~\Sigma$ let $\mathop{\rm iocr}\nolimits_{\Sigma}(G)$ denote the fewest number of pairs of independent edges that cross oddly in a drawing of $G$ on $\Sigma$, while $\mathop{\rm cr}\nolimits_{\Sigma}(G)$ denotes the crossing number on $\Sigma$. Quadratic surface crossing-number conjecture. For $\Sigma$ equal to the projective plane or the torus, one has

$$\mathop{\rm cr}\nolimits_{\Sigma}(G) \leq \binom{2\mathop{\rm iocr}\nolimits_{\Sigma}(G)}{2}.$$

The corresponding inequality is known for the plane, but analogous bounds are not known for other surfaces, including the projective plane and torus, so this proposes a concrete quadratic bound in those cases.