Cairns–Nikolayevsky thrackle conjecture for orientable surfaces

Let $S_g$ be a compact orientable surface of genus $g>0$, and let $G$ be a graph with $n$ vertices and $m$ edges. Cairns–Nikolayevsky's conjecture. If $G$ can be thrackled on $S_g$, then

$$m\le n+2g.$$

Odd cycles and recursive constructions provide examples with $n+2g$ edges, motivating this proposed maximum. The paper states that this conjecture is disproved by constructions with substantially more edges, so the claim is not open.