Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:

- The entries of $A$ are $\in \{0, 1\}$.
- For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are never either all non-negative or all non-positive (i.e. there is a positive entry and a negative entry in $u - v$).
- $\sum_i x_i = 0$.
- The entries of $Ax$ are all non-negative with at least one entry being strictly positive.

Edit: It turns out this is not true via a concrete counterexample found by my collaborator. I would list it, but it's rather large.