Cube-ideal set conjecture for bounded non-agreement certificates

Let $S\subseteq\{0,1\}^n$. Call $S$ cube-ideal if every facet of $\operatorname{conv}(S)$ is defined by $x_i\geq0$, $x_i\leq1$, or an inequality

$$\sum_{i\in I}x_i+\sum_{j\in J}(1-x_j)\geq1$$

for disjoint $I,J\subseteq[n]$. Points agree on a coordinate if they all lie in a common coordinate hyperplane $x_i=a$ for some $i\in[n]$ and $a\in\{0,1\}$.

Cube-ideal set conjecture. There exists an integer $k\geq4$ such that for every cube-ideal set, either all the points agree on a coordinate, or there is a subset of at most $k$ points that do not agree on a coordinate.

This is the cuboid reformulation of the main non-idealness conjecture. It asks for a uniformly bounded certificate of failure of coordinate agreement in cube-ideal sets.