Vanishing conjecture for random graph neighborhood-complex cohomology

Let $G(n,p)$ be the binomial random graph, and let $\mathcal{N}[G(n,p)]$ denote its neighborhood complex. For an integer $k\geq 0$, let $\widetilde{H}^{k+1}(\mathcal{N}[G(n,p)])$ denote its reduced cohomology group.

Cohomology-vanishing conjecture. If

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

then, asymptotically almost surely,

$$\widetilde{H}^{k+1}(\mathcal{N}[G(n,p)])=0.$$

This is the proposed extension of the preceding result to the case $s=0$. It is stated as further research, and the source gives no resolution.