The coordinate-deletion conjecture for powerful sets

Let $S\subseteq\mathbb{F}_2^n$ be a powerful set with $2^{n-1}$ elements, where $n\geq2$. Deleting a coordinate means removing that coordinate from every vector; the resulting vectors are distinguishable when they form the set $\mathbb{F}_2^{n-1}$.

Coordinate-deletion conjecture. If $S$ is a powerful set, then one can find a coordinate such that deleting this coordinate from all the elements of $S$ yields the set $\mathbb{F}_2^{n-1}$, so that all the new vectors are distinguishable.

This conjecture concerns the structure of powerful sets of exactly half the size of the ambient binary cube. The supplied text gives no indication that the claim is resolved, so it remains open here.