The all-codimension stability conjecture for Hamming-cube subcubes

Let $Q_n$ be the $n$-dimensional Hamming cube with vertex set $V$. For a set $A\subseteq V$, let $\mu(A)=|A|/2^n$, let $\nabla(A,B)$ denote the set of edges between $A$ and $B$, and let $\partial(a)$ be the minimum vertex boundary among sets of size $a$. Suppose $(A,B,W)$ is a partition of $V$. For $k\in\mathbb P$, the statement below refers to the conclusion that there is a codimension-$k$ subcube $C$ with $\mu(C\mathbin{\Delta}A)=O(\epsilon)$ under the hypotheses of Theorem

$$, with the theorem's corresponding edge and measure conditions. **All-codimension stability conjecture.** The statement in Theorem$$

holds for all $k\in\mathbb P$, even with $n^\beta$ replaced by $2^n/\partial(|A|)$.

The conjecture extends the proved stability theorem from $k\in\{1,2\}$ to every positive codimension and replaces the stated control on the exceptional set by a quantity involving the minimum vertex boundary. The source gives no resolution of this extension.