Very short sets conjecture for the internal zonotopal space

Let $X$ be a finite list of vectors of rank $n$. Define the set of very short $X$-sets by

$$S_-(X)=\{Y\subset X:\operatorname{rank}(X\setminus(Y\cup\{x\}))=n\text{ for every }x\in X\setminus Y\}.$$

Let ${\cal P}_-(X)$ be the internal ${\cal P}$-space. Very short sets conjecture. For every $X$,

$${\cal P}_-(X)=\operatorname{span}\{p_Y:Y\in S_-(X)\}.$$

The right-hand side is the earlier formulation of the internal ${\cal P}$-space and is already known from the supplied context to be a subspace of ${\cal P}_-(X)$; the conjecture asserts the reverse inclusion, so that the two variants coincide.