Janson's asymptotic Gaussianity conjecture for the number of vertices

Let $\mu \in \mathcal{P}(\mathbb{N}_2)$ be the law governing the edge-exchangeable graph, and let $V_n$ be its vertex set after $n$ edges have been sampled. Assume that

$$\mathbb{V}[|V_n|] \longrightarrow \infty.$$

Janson's asymptotic Gaussianity conjecture. Under this assumption, the number of vertices $|V_n|$ is asymptotically Gaussian.

This conjecture concerns the central-limit behaviour of vertex counts in edge-exchangeable graphs. The supplied text identifies it as Problem 6.9 of Janson's work; no resolution is given here.