A counterexample to a conjecture of Cerf on second-order pivots

For an increasing subset $A$ of $\{0,1\}^n$, Cerf considers the number $\phi_2(A)$ of pairs $(\omega,i)$ with $\omega\in A$ and $i$ a positive pivot of order at most two at $\omega$, and the analogous count $\phi_{2\setminus1}(A)$ of pivots of order exactly two. He conjectured that every maximizer of either functional contains all configurations with at least $n/2+1$ ones and no configuration with fewer than $n/2-2$ ones, which would give $\phi_2(A)\le 6\sqrt n\,2^n$. We show that the conjecture fails. For $n=10^{14}$, and more generally for every $n=h^2$ with $10\mid h$ and $h>2{,}592{,}000$, there is an increasing set with $\phi_{2\setminus1}(A)\ge 2^n n/259200$, while every increasing set inside the conjectured band has $\phi_2(A)\le 10\sqrt n\,2^n$. Consequently no maximizer of either functional lies in the band. The set is a random monotone CNF of the kind used by Talagrand, and the proof is a first and second moment count over the clauses.