The universal asymptotic unlabelled 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. Universal asymptotic unlabelled fragment-size conjecture. There is a constant $c$ such that, for each bridge-addable and decomposable graph class $\mathcal A$,

$$\limsup_{n\to\infty}\mathbb E[\operatorname{frag}(\tilde{R}_n)]\leq c.$$

The source describes this as a speculative final conjecture. It differs from the preceding assertion by proposing one universal constant rather than a constant depending on $\mathcal A$.