A tame sequence of transitive Boolean functions

Given a sequence of Boolean functions $(f_n{)}_{n\ge 1}$, $f_n \colon { 0,1 }^{n} \to { 0,1 }$,andasequence$(X^{(n)}){n\geq 1} $ofcontinuoustime$p_n $-biasedrandomwalks$ X^{(n)} = (X_t^{(n)}){t \geq 0}$on$ { 0,1 }^{n}$,let$ C_n $bethe(random)numberoftimesin$(0,1) $ at which the process $(f_n(X_t){)}_{t\ge 0}$ changes its value. In \cite{js2006}, the authors conjectured that if $(f_n{)}_{n\ge 1}$ is non-degenerate, transitive and satisfies ${\lim }_{n\to \mathrm{\infty }}\mathbb{E}[C_n]=\mathrm{\infty }$, then $ (C_n)_{n \geq 1} $ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.