Hliněný's higher-genus non-orientable cover conjecture

Let $\Sigma$ be a connected compact non-orientable surface without boundary. A finite $\Sigma$-cover is a finite cover of a connected graph that embeds in $\Sigma$. Hliněný's conjecture. A connected graph has a finite $\Sigma$-cover if and only if it is embeddable in $\Sigma$. Hliněný proposed this as a higher-genus extension of Negami's conjecture; it remains open, while the paper discusses partial progress and obstructions to analogous statements for orientable surfaces.