Typical string graph convergence conjecture

Let $\mathcal{SG}_n$ and $\mathcal{SG}^L_n$ denote, respectively, the classes of unlabeled and labeled string graphs on $n$ vertices, and let $W^4_{1/2}$ be the corresponding graph limit. For each $n$, choose $G_n$ uniformly from the relevant class. Typical string graph convergence conjecture. For uniformly random unlabeled $G_n\in\mathcal{SG}_n$, $G_n\overset{\mathrm{p}}{\longrightarrow}W^4_{1/2}$; the same conclusion holds for uniformly random labeled $G_n\in\mathcal{SG}^L_n$. This specifies the conjectured limiting graph limit within the one-parameter family of possible string-graph limits; the preceding discussion gives motivation for the value $1/2$, but no resolution is stated.