Connectivity conjecture for random graph neighborhood complexes

Let $G(n,p)$ be the binomial random graph, and let $\mathcal{N}[G(n,p)]$ denote its neighborhood complex. For integers $k\geq s\geq 1$, write $o(n^s)$ with respect to $n\to\infty$.

Connectivity conjecture. If

$$\binom{n}{k+2}(1-p^{k+2})^{n-k-2}=o(n^s),$$

then $\mathcal{N}[G(n,p)]$ is asymptotically almost surely $(k-s+1)$-connected.

This would strengthen the cited connectivity result for the case $s=1$; the conjecture is presented as a direction for further research, and no resolution is given in the source.