Number of Eulerian orientations for Benjamini--Schramm convergent graph sequences

For a graph $G$ let $\varepsilon(G)$ denote the number of Eulerian orientations, and $v(G)$ denote the number of vertices of $G$. We show that if $(G_n)_n$ is a sequence of Eulerian graphs that are convergent in Benjamini--Schramm sense, then $\lim\limits_{n\to \infty}\frac{1}{v(G_n)}\ln \varepsilon(G_n)$ is convergent.