The cover inequality for Eulerian orientations

Let $G$ be an Eulerian graph, and let $H$ be a $k$-cover of $G$. Here, $\varepsilon(G)$ denotes the number of Eulerian orientations of $G$.

Cover inequality. Then

$$\varepsilon(G)^k\geq \varepsilon(H).$$

This is posed as an open problem concerning how the number of Eulerian orientations behaves under graph covers.