Directed KKL inequality

Let $f\colon\{0,1\}^n\to\{0,1\}$ be a Boolean function. Define its distance to monotonicity by

$$\varepsilon(f):=\min_{g\ \mathrm{monotone}}\Pr_{x\sim\{0,1\}^n}[f(x)\neq g(x)].$$

For $i\in[n]$, define the negative influence of coordinate $i$ on $f$ by

$$\operatorname{Inf}^{-}_i[f]:=\#\{x:f(x)>f(x^{\oplus i})\text{ and }x\preccurlyeq x^{\oplus i}\}\cdot\frac{1}{2^{n-1}}.$$

Directed KKL inequality. Given a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$, there exists $i\in[n]$ such that

$$\operatorname{Inf}^{-}_i[f]\geq\Omega\left(\varepsilon(f)\cdot\frac{\log n}{n}\right).$$

This is the natural directed analogue of the Kahn–Kalai–Linial inequality, with distance to monotonicity replacing variance and negative influences replacing ordinary influences. The supplied material does not establish whether the inequality is proved or remains open.