Kahn–Kalai large-subcube conjecture for monotone cube sets

Let $A\subset\{0,1\}^n$ be monotone increasing, with measure $p=|A|/2^n\leq 1/2$, and let $\partial A$ denote its edge-boundary. A subcube is obtained by fixing coordinates; its codimension is the number of fixed coordinates.

Kahn–Kalai conjecture. For every $L>0$, there exist $L'>0$ and $\delta>0$ such that if

$$|\partial A|\leq L|A|\log_2(2^n/|A|),$$

then there is a subcube $C\subset\{0,1\}^n$ of codimension at most $L'\log_2(1/p)$, with all fixed coordinates equal to $1$, satisfying

$$\frac{|A\cap C|}{|C|}\geq(1+\delta)p.$$

The conjecture predicts that a monotone set with boundary within a constant factor of optimal must have increased density on a relatively low-codimension upper subcube. The source leaves it open and proposes a non-monotone extension.