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

For a graph $G$ let $\epsilon (G)$ 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 $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{v(G_n)}\ln \epsilon (G_n)$ is convergent.