When are sequences of Boolean functions tame?

In \cite{js2006}, Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions $(f_n)_{n \geq 1}$ with $\lim_{n \to \infty} I(f_n)= \infty$ could be tame (with respect to some $(p_n)_{n \geq 1}$). In a companion paper \cite{f}, the author showed that this conjecture in its full generality is false, by providing a counter-example for the case when, at the same time, $\lim_{n \to \infty} np_n = \infty$ and $\lim_{n \to \infty} n^\alpha p_n = 0$ for some $\alpha \in (0,1 ).$ In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence $(p_n)_{n \geq 1}$ is bounded away from zero and one.