Dimension-stability conjecture for exact hypercube covering numbers

For $B\subset\{0,1\}^k$ and an integer $n\geq k$, let $\operatorname{ec}(B)$ be the minimum number of hyperplanes whose union intersects the Boolean cube precisely in $B$. Dimension-stability conjecture.

$$\operatorname{ec}\bigl(\{0,1\}^{n}\setminus(B\times\{0\}^{n-k})\bigr)=n-k+\operatorname{ec}\bigl(\{0,1\}^{k}\setminus B\bigr).$$

The right-hand side is an immediate upper bound obtained by extending a cover in $k$ dimensions and adding the $n-k$ coordinate hyperplanes. The conjecture asks whether embedding the problem in higher dimensions can ever improve this bound; the source gives no resolution.