Kalai–Keller–Mossel correlation conjecture for increasing balanced families

Let $\Omega_n=\{0,1\}^n$ with the uniform measure $\mu$. A family $\mathcal{A}\subseteq\Omega_n$ is increasing if $x\leq y$ coordinate-wise implies $x\in\mathcal{A}$ whenever $y\in\mathcal{A}$, and balanced if $\mu(\mathcal{A})=1/2$. For a family $\mathcal{F}$, write $\mathbf{1}_{\mathcal{F}}$ for its indicator, and let $\operatorname{Cov}_\mu$ denote covariance under $\mu$. An increasing linear-threshold family is a family of the form

$$\mathcal{B}=\left\{x\in\Omega_n:\sum_{i=1}^n a_i x_i>t\right\},$$

where $a_i\geq 0$ and $t\in\mathbb{R}$.

Kalai–Keller–Mossel correlation conjecture. For every increasing and balanced family $\mathcal{A}\subseteq\Omega_n$, there exists an increasing linear-threshold family $\mathcal{B}$ with non-negative weights such that

$$\operatorname{Cov}_\mu(\mathbf{1}_{\mathcal{A}},\mathbf{1}_{\mathcal{B}})\geq c\,\frac{\log n}{\sqrt{n}},$$

where $c>0$ is a universal constant.

The paper states that its second main theorem proves this conjecture, by finding a suitable coordinate projection or the majority function; thus the conjecture is solved.