The unlabelled bridge-addable connectivity conjecture

Let $\mathcal A$ be a class of graphs, let $\tilde{\mathcal A}_n$ be the set of unlabelled graphs in $\mathcal A$ on $n$ vertices, and let $\tilde{R}_n \in_u \tilde{\mathcal A}$ denote a graph sampled uniformly from $\tilde{\mathcal A}_n$. A graph class is bridge-addable if adding an edge between vertices in distinct components preserves membership. Unlabelled bridge-addable connectivity conjecture. There is a $\delta>0$ such that, if $\mathcal A$ is bridge-addable, then

$$\mathbb P(\tilde{R}_n \text{ is connected})\geq\delta$$

for each $n$. This asks for a uniform positive lower bound on connectivity in the unlabelled setting; the source notes that little was known about this case and refers to a theorem below that bears on the question.