The transfinite columns property conjecture

Let $A$ be an $\omega\times\omega$ matrix with entries from $\mathbb Z$, and assume that $A$ is partition regular. A matrix has bounded row sums when the sums of the absolute values of the entries in its rows are bounded. Writing the columns as $\langle\vec c_i\rangle_{i<\omega}$, $A$ has the transfinite columns property if there are a countable ordinal $\mu$ and a partition $\langle I_\sigma\rangle_{\sigma<\mu}$ of $\omega$ such that

$$\sum_{i\in I_0}\vec c_i=\vec 0$$

and, for every $t\in\mu\setminus\{0\}$, $\sum_{i\in I_t}\vec c_i$ is a rational linear combination of $\{\vec c_i:i\in\bigcup_{j<t}I_j\}$.

Transfinite columns property conjecture. If $A$ has bounded row sums, then $A$ has the transfinite columns property.

This is proposed as a weaker substitute for the bounded-row-sums columns property conjecture. The paper notes that matrices can require arbitrarily large countable ordinals in this transfinite condition, and leaves the proposed statement open.