The coloop-extension conjecture for powerful sets

Let $T\subseteq\mathbb{F}_2^n$ be a powerful set, meaning that for every subset of coordinates, the number of vectors in $T$ that are zero on those coordinates is a power of $2$. A vector has weight $1$ if exactly one of its coordinates is nonzero. A coloop extension of a set is obtained by adjoining a new coordinate so that each vector of a set $S$ occurs with both possible values in that coordinate.

Coloop-extension conjecture. If $T$ is a powerful set with at least one vector of weight $1$, then $T$ is a coloop extension of some powerful set $S$.

The preceding observation shows that every coloop extension of a powerful set has a vector of weight $1$; the conjecture asserts the converse. The supplied text does not indicate whether this converse has been proved or disproved.