The high-genus surface-cover extension of Negami's conjecture

Let $\Sigma$ be a connected compact surface without boundary and let $g$ be its Euler genus. A finite $\Sigma$-cover is a finite cover of a connected graph that embeds in $\Sigma$. Surface-cover extension. A connected graph has a finite $\Sigma$-cover if and only if it is embeddable in a surface finitely covered by $\Sigma$. This conjecture unifies the non-orientable and orientable formulations above and remains open; the paper proves substantial evidence, including bounded Euler genus for graphs with finite covers embeddable in a fixed surface.