Kantor's nilpotency conjecture for random subgroups of symmetric groups

Let $\mathrm{S}_n$ be the symmetric group on $n$ points. A random subgroup of $\mathrm{S}_n$ means a subgroup chosen uniformly from the finite set of all subgroups of $\mathrm{S}_n$. Kantor's conjecture. As $n$ tends to infinity, a random subgroup of $\mathrm{S}_n$ is nilpotent, equivalently, the proportion of nilpotent subgroups among all subgroups of $\mathrm{S}_n$ tends to $1$.

The source attributes this formulation to Kantor. The supplied material does not establish whether the conjecture is resolved; the paper proves upper bounds for nilpotent subgroups and also notes that for infinitely many $n$ the probability of nilpotency is bounded away from $1$, which is relevant to assessing the conjecture's status.