The fixed-error 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 $V$. Write $abla(A,B)=\{(x,y):x\in A,y\in B\}$ for the edges between $A$ and $B$, and let $\mu(A)=|A|/2^n$.

Fixed-error hybrid isoperimetric conjecture. There is a fixed constant $K$ such that, whenever $\mu(A)=1/2$,

$$|\nabla(A,B)|+K\sqrt{n}\,|W|\geq 2^{n-1}.$$

This is a proposed hybrid form of the edge-isoperimetric inequality, quantifying how a small exceptional part $W$ affects separation of two parts of the cube. The source gives no resolution of the conjecture.