The unlabelled bridge-addable fragment-size conjecture

Let $\mathcal A$ be a class of graphs that is bridge-addable, meaning that adding an edge between vertices in distinct components preserves membership, and decomposable, meaning that a graph belongs to $\mathcal A$ if and only if each of its components does. For a graph $G$, define its fragment size by

$$\operatorname{frag}(G)=|V(G)|-\max\{|V(C)|:C\text{ is a component of }G\}.$$

Let $\tilde{R}_n \in_u\tilde{\mathcal A}$ be sampled uniformly from the unlabelled graphs in $\mathcal A$ on $n$ vertices. Unlabelled bridge-addable fragment-size conjecture. For each such graph class $\mathcal A$, there is a constant $c_{\mathcal A}$ such that, for each positive integer $n$,

$$\mathbb E[\operatorname{frag}(\tilde{R}_n)]\leq c_{\mathcal A}.$$

The labelled analogue has a uniform expectation bound, while the source gives an unlabelled example showing that the expectation can grow linearly for a particular class; the conjectured bound remains open in general.