Typical outer-string graph edge-density and degree-distribution conjecture

For a graph $G$ on $n$ vertices, let $X_n$ be the degree of a uniformly random vertex of $G$. Let $\mathcal{OSG}_n$ and $\mathcal{OSG}^L_n$ denote the classes of unlabeled and labeled outer-string graphs on $n$ vertices. Typical outer-string graph edge-density and degree-distribution conjecture. If $G_n$ is uniformly random in $\mathcal{OSG}_n$, then

$$\frac{e(G_n)}{\binom{n}{2}}\overset{\mathrm{p}}{\longrightarrow}\frac{11}{18}.$$

Furthermore, $X_n/n$ converges in distribution to a random variable $Y$ such that $Y\in\{1/2,2/3\}$ almost surely, with

$$\mathbb{P}(Y=1/2)=\frac13,\qquad \mathbb{P}(Y=2/3)=\frac23.$$

The same conclusions hold when $G_n$ is uniformly random in $\mathcal{OSG}^L_n$. These conjectured limits are the outer-string analogue of the proposed string-graph statistics and predict a bimodal degree distribution; the source states no resolution.