Regularity-free simple elevation theorem for covers

Let $S$ be a surface and let $S_1$ and $S_2$ be covers of $S$. A closed curve on $S$ has a simple elevation to a cover if one of its elevations is a simple closed curve. Regularity-free simple elevation conjecture. Theorem on simple elevations for regular covers holds without the assumption that the covers are regular. This conjecture would address the first step in generalizing the analysis of Sunada covers: if two covers are not equivalent, one should be able to find a closed curve on the base with different simple-elevation behavior in the two covers. The result is open in the source, and its precise content depends on the omitted statement of Theorem thm:regular.