Random-graph conjecture on bounded expected nontrivial invariant factors

Let $\EuScript{G}(n,p)$ be a random simple undirected graph with $n$ vertices and edge-probability $0\leq p\leq 1$. For $c>1$, assume $c\log(n)/n<p(n)<1-o(\log(n)/n)$. Bounded-expectation conjecture. As $n\to\infty$, the expected value of $\nu(\EuScript{G}(n,p))$ remains bounded. This is presented as a weaker and potentially more accessible form of the preceding cyclicity conjecture. The source gives no resolution and only motivates it through the preceding discussion of critical groups and Smith normal forms.