  Population protocols are a model of computation in which an arbitrary number
of anonymous finite-memory agents are interacting in order to decide by stable
consensus a predicate. In this paper, we focus on the counting predicates that
asks, given an initial configuration, whether the number of agents in some
initial state $i$ is at least $n$. In 2018, Blondin, Esparza, and Jaax shown
that with a fix number of leaders and interaction-width, there exists
infinitely many $n$ for which the counting predicate is stably computable by a
protocol with at most $O(\log\log(n))$ states. We provide in this paper a
matching lower-bound (up to a square root) that improves the
inverse-Ackermannian lower-bound presented at PODC in 2021.
