Kahn and Kalai's dense-subcube conjecture

Let $A\subset\{0,1\}^n$ be monotone increasing, with measure $\alpha=|A|/2^n\leq 1/2$. Its edge-boundary is denoted by $\partial A$. Kahn and Kalai's 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 measure at least $\alpha^{L'}$, with all fixed coordinates equal to $1$, such that

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

The hypothesis requires the boundary to be within a constant factor of the edge-isoperimetric minimum, while the conclusion finds a fairly large subcube on which the density increases by a constant factor. The source gives no resolution.