The general CNF–DNF correlation inequality for random points in a hypercube

Let $U_1,\ldots,U_k,U_{k+1},\ldots,U_{2k},U_{2k+1},\ldots,U_n$ be independent and identically distributed random variables with the uniform distribution on $(0,1)$. Let $F_1,\ldots,F_T\subset [k]\times\{2k+1,\ldots,n\}$ and $H_1,\ldots,H_T\subset\{k+1,\ldots,2k\}\times\{2k+1,\ldots,n\}$, where $T\geq 1$, $|F_t|=|H_t|=m\leq k$ for every $t\in[T]$, and

$$\sum_{j=1}^{m}\mathbf{1}_{\{(i,j)\in F_t\}}\leq 1\quad\text{for every }i\in[k],\ t\in[T],$$

with the analogous condition

$$\sum_{j=1}^{m}\mathbf{1}_{\{(i,j)\in H_t\}}\leq 1\quad\text{for every }i\in\{k+1,\ldots,2k\},\ t\in[T].$$

General CNF–DNF correlation conjecture. Under these assumptions, the inequality referred to in the source as the CNF–DNF inequality holds.