In [10], Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions (f(n))(n >= 1) with lim(n ->infinity) I(f(n)) = infinity could be tame (with respect to some (p(n))(n >= 1)). In a companion paper [5], 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 ->infinity) np(n) = infinity and lim(n ->infinity) n(alpha)p(n) = 0 for some alpha is an element of (0, 1). In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence (p(n))(n >= 1) is bounded away from zero and one.