The cactus-graph probability conjecture for sparse random graphs

Let $A(n,m)$ denote the random graph with $n$ vertices and $m$ edges. For $1/2<\alpha<1$, set $m=\alpha n$. Cactus-graph probability conjecture. The probability that $A(n,m)$ is a cactus graph is bounded away from $0$ and $1$. This conjecture asks how the probability of being a cactus graph behaves in the linear-edge regime; the supplied text gives no resolution, so its status remains open.