The maximal hybrid isoperimetric conjecture for the Hamming cube

Let $Q_n$ be the $n$-dimensional Hamming cube with vertex set $V$, and let $(A,B,W)$ be a partition of $V$. For $a=|A|$, define

$$\partial(a)=\min\{|\partial A|:|A|=a\},$$

and let $\nabla(a)$ be defined analogously using the edge boundary. Write $\nabla(A,B)=\{(x,y):x\in A,y\in B\}$.

Maximal hybrid isoperimetric conjecture. If $|A|=a$, then

$$\frac{|\nabla(A,B)|}{\nabla(a)}+\frac{|W|}{\partial(a)}\geq 1.$$

This conjecture would give a general tradeoff between the edges joining $A$ to $B$ and the exceptional set $W$, extending the usual vertex- and edge-isoperimetric bounds. The source gives no resolution of the conjecture.