The linear expected-face conjecture for random orientable embeddings

Let $G$ be a simple graph of order $n$. Select an orientable embedding of $G$ uniformly at random, and let $F$ denote its number of faces. Linear expected-face conjecture. The expected number of faces satisfies

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

Stahl proved the upper bound $\mathbb{E}[F]\leq n\log n$, while many bounded-degree graph families with linearly many short cycles have $\mathbb{E}[F]=\Theta(n)$. No examples with superlinear expected numbers of faces are known, so the conjecture asks whether the logarithmic factor can always be removed for simple graphs.