The threshold-exponent conjecture for K_d-completion

Consider the $K_d$-completion dynamics on $[n]$, in which copies of $K_d$ missing one edge are iteratively completed, with initial graph $G_0$ having edges $i\leftrightarrow j$ whenever $|i-j|\le d-2$. Let $p_c$ denote the critical probability for saturation. $K_d$-completion threshold-exponent conjecture. There exists a power $\gamma=\gamma(d)>0$ such that, for large $n$, $p_c$ lies between two constant multiples of

$$(\log\log n)^{\gamma}(\log n)^{-1/(d-1)}.$$

This predicts the threshold scale for nucleation and saturation in the $K_d$-completion dynamics; the paper presents the claim as an unresolved conjecture motivated by simulations and known results for the unpolluted process.