Alon et al.'s conjecture on iterativity and strong adversaries

Let a graph be evaluated under an opinion-forming model with initially selected experts, binary labels, and an adversary that chooses the experts' labels. In the iterative propagation process, unlabeled vertices repeatedly adopt the majority label among their labeled neighbours until all vertices are labeled; the corresponding non-iterative propagation process uses only the initial dissemination round. A graph is robust against an adversary if that adversary cannot make a majority of the graph believe the false opinion. Alon et al.'s conjecture. In the case of a strong adversary, iterative propagation can never harm the adversary. Equivalently, there is no graph that is robust against the iterative strong adversary but not robust against the non-iterative strong adversary. Thus, iterativity should not decrease the adversary's ability to influence the final majority; the source presents this as a conjecture attributed to Alon et al., while contrasting it with an example showing that the analogous claim fails for a weak adversary.