The unlabelled bridge-addable connectivity lower-bound 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. Let $\tilde{\mathcal A}_n$ be the set of unlabelled graphs in $\mathcal A$ on $n$ vertices, and let $\tilde{R}_n \in_u \tilde{\mathcal A}$ be sampled uniformly from this set. Let $\tau$ be the constant determined by

$$\frac{|\tilde{\mathcal T}_n|}{|\tilde{\mathcal F}_n|}\longrightarrow\tau\approx e^{-0.5226}\approx0.5930,$$

where $\mathcal T$ and $\mathcal F$ are the classes of trees and forests, respectively. Unlabelled bridge-addable connectivity lower-bound conjecture. If $\mathcal A$ is bridge-addable and decomposable, then

$$\liminf_{n\to\infty}\mathbb P(\tilde{R}_n\text{ is connected})\geq\tau.$$

This is the unlabelled counterpart of the labelled forest lower bound, which the source says was proved by Chapuy and Perarnau; the unlabelled assertion is described as more speculative.