Linear thrackle-genus conjecture for complete bipartite graphs $K_{2,n}$

Let $\operatorname{tg}(G)$ denote the minimum genus of an orientable surface on which the graph $G$ can be thrackled. Linear thrackle-genus conjecture.

$$\operatorname{tg}(K_{2,n})=\Theta(n).$$

The source has already established nontrivial bounds, including a lower bound of order $n^{1/3}$, and says that the authors slightly lean toward the linear upper bound. The conjecture remains open.