Consistency of the positive terraced cube relation at the first infinite cardinal

Determine whether the positive terraced cube relation \(\left(\begin{smallmatrix}\aleph_2\\\aleph_1\\\aleph_0\end{smallmatrix}\right)\rightarrow\left(\begin{smallmatrix}\aleph_2\\\aleph_1\\\aleph_0\end{smallmatrix}\right)\) is consistent with ZFC.

Background

The paper studies strong cube polarized relations whose three domain cardinals and corresponding monochromatic target cardinals are identical. It proves in ZFC that the $\lambda$-terraced cube relation fails for every uncountable cardinal $\lambda$, but the case $\lambda=\aleph_0$ is not settled.

For $\lambda=\aleph_0$, the paper notes that a positive relation would require substantial consistency strength, including at least one Woodin cardinal, and that forcing it would require $2^{\aleph_0}\geq\aleph_3$. The remaining consistency question is therefore a specific unresolved issue about the first infinite terraced cube.