The linear expected-face conjecture for multigraphs

Let $G$ be a multigraph on $n$ vertices, and let $\mu$ be its maximum edge-multiplicity. Choose an orientable embedding of $G$ uniformly at random. Linear multigraph expected-face conjecture. The expected number of faces is

$$O\bigl(n\log(2\mu)\bigr).$$

This adjusts the simple-graph conjecture to account for parallel edges. The paper presents it after proving bounds for simple graphs and does not establish it in full; its status is open.