Typical distance lower-bound conjecture for multipartite random graphs

Let $G_n$ be the random graph and let $o_1,o_2$ be the two vertices whose distance is considered. Assume the hypotheses of Theorem. Then typical distance lower-bound conjecture. There exists $K>0$ such that

$$\liminf_{n \to \infty}\mathbb{P}\left(d_{G_n}(o_1,o_2)>K \log n\right)>0.$$

This conjecture asserts a logarithmic lower bound on typical distances even when some pairs of communities are BAD; the paper explains that it cannot prove the corresponding result without assuming that all pairs are GOOD, and expects the presence of BAD pairs to reduce the number of edges and therefore increase distances.