Kahle's homology vanishing conjecture for random clique complexes

Let $G=G_{n,p}$ be the Erdős–Rényi random graph, and let $X(G)$ be its clique complex, whose faces are the vertex sets of cliques of $G$. Let $\Gamma$ be either $\mathbb{Z}$ or a field, let $k$ be a positive integer, and let $H_k(X(G),\Gamma)$ denote the $k$th homology group. An event holds w.h.p. if its probability tends to $1$ as $n\to\infty$.

Kahle's homology vanishing conjecture. For each positive integer $k$ and each $\varepsilon>0$, if

$$p>(1+\varepsilon)\left[(1+k/2)(\log n/n)\right]^{1/(k+1)},$$

then w.h.p.

$$H_k(X(G),\Gamma)=0.$$

The source attributes this conjecture to M. Kahle and notes that it was proved there for $\Gamma=\mathbb{Q}$. The conjecture concerns the threshold above which the $k$th homology of a random clique complex vanishes.