The linear expected-face conjecture for simple graphs

Let $G$ be a simple graph on $n$ vertices, and choose an orientable embedding of $G$ uniformly at random. Write $F_G$ for the number of faces of the resulting embedding. Linear expected-face conjecture. For every such graph $G$, the expected number of faces is

$$\mathbb{E}[F_G]=O(n).$$

The paper establishes logarithmic upper bounds and gives linear lower-bound constructions, but does not find a family with superlinear expected numbers of faces; the conjecture remains open.