Critical probability conjecture for bootstrap percolation on Hamming graphs

Let $\square_{i=1}^n K_k$ be the Cartesian product of $n$ copies of the complete graph $K_k$, and let $p_c(G,r)$ denote the critical probability for $r$-neighbor bootstrap percolation on a graph $G$. For arbitrary $r\geq 3$ and $k\geq 2$, the critical probability conjecture asserts that

$$-\log_k p_c\left(\square_{i=1}^n K_k,r\right)=\Theta\left(n^{1/2^{r-1}}\right).$$

This generalizes the conjecture of Balogh, Bollobás and Morris for the hypercube, corresponding to $k=2$. The statement remains open, including the previously open hypercube case for arbitrary $r\geq 3$.