The orientable higher-genus extension of Negami's conjecture

Let $\Sigma$ be an orientable surface of Euler genus $g$, and let a finite $\Sigma$-cover mean a finite cover of a connected graph that embeds in $\Sigma$. Orientable higher-genus conjecture. A connected graph has a finite $\Sigma$-cover if and only if it is embeddable in $\Sigma$ or embeddable in the non-orientable surface of genus $g/2+1$. This is motivated by the fact that the relevant orientable surface covers itself and the corresponding non-orientable surface; the conjecture is presented as a natural analogue and remains open.