  Majority dynamics on the binomial Erd\H{o}s-R\'enyi graph $\mathsf{G}(n,p)$
with $p=\lambda/\sqrt{n}$ is studied. In this process, each vertex has a state
in $\{0,1\}$ and at each round, every vertex adopts the state of the majority
of its neighbors, retaining its state in the case of a tie. It was conjectured
by Benjamini et al. and proved by Fountoulakis et al. that this process reaches
unanimity with high probability in at most four rounds. By adding some extra
randomness and allowing the underlying graph to be drawn anew in each
communication round, we improve on their result and prove that this process
reaches consensus in only three communication rounds with probability
approaching $1$ as $n$ grows to infinity. We also provide a converse result,
showing that three rounds are not only sufficient, but also necessary.
