The conjectured sharp threshold for connectivity and Hamiltonicity in random intersection graphs

Let $\alpha<1$, let

$$p=\frac{\ln n+\omega}{m},$$

and suppose that $\omega\to\infty$. Then, with high probability, the random intersection graph $\mathcal{G}(n,m,p)$ is $k$-connected for any constant $k$ and contains a Hamilton cycle.

Connectivity and Hamiltonicity conjecture. Under these assumptions, $\mathcal{G}(n,m,p)$ is $k$-connected for every fixed constant $k$ and contains a Hamilton cycle with high probability.

The conjecture is motivated by the minimum-degree phenomenon and proposes a tightening of the threshold function in the cited theorem for the case $\alpha<1$. The supplied text gives no resolution of this conjecture.