The chain-of-triangles bound for expected faces

Let $G$ be a simple graph of order $n$, and let $F$ denote the number of faces of a uniformly random orientable embedding of $G$. Chain-of-triangles conjecture. For every such graph,

$$\mathbb{E}[F]\leq \frac{1}{3}n+1.$$

A chain of triangles connected by cutedges attains $\mathbb{E}[F]=\frac{1}{3}n+1$, so the conjecture proposes that this construction is extremal. The source first establishes the general upper bound $\mathbb{E}[F]<\frac{\pi^2}{6}n$ for simple graphs, leaving the sharper constant conjectured here.