The modified transitive Boolean-function tameness conjecture

Let $(p_n)_{n\geq 1}$ be a sequence of biases and let $(f_n)_{n\geq 1}$ be Boolean functions, with $C_n$ the number of changes of $f_n(X_t^{(n)})$ during $(0,1)$. A sequence $(f_n)$ is transitive when each function is invariant under a coordinate-permutation group acting transitively on the coordinates, and non-degenerate with respect to $(p_n)$ when its output probabilities remain bounded away from $0$ and $1$. It is tame when $(C_n)_{n\geq 1}$ is tight. Modified transitive Boolean-function tameness conjecture. If $(p_n)_{n\geq 1}$ satisfies

$$\lim_{n\to\infty}np_n^r=\infty$$

for every $r>0$, and $(f_n)_{n\geq 1}$ is transitive and non-degenerate with respect to $(p_n)_{n\geq 1}$, then $(f_n)_{n\geq 1}$ is not tame. This modification removes the expected-change hypothesis from the earlier conjecture and imposes a growth condition on the bias sequence; the supplied text presents it as a suggested modified conjecture, with no resolution given.