The non-cactus conjecture for near-linear sparse random graphs

Let $A(n,m)$ denote the random graph with $n$ vertices and $m$ edges, and write $m=n+t$ with $t=o(n)$. Non-cactus conjecture. With high probability, $A(n,m)$ is not a cactus graph. This conjecture concerns the regime just above the tree threshold; the supplied text gives no resolution, so its status remains open.