 Solution

No. The incompatibility part of the Milliken–Taylor theorem provides a finite coloring of $\mathbb{N}$ for which the systems associated with the nonproportional compressed coefficient vectors $(1)$ and $(1,2,3)$ cannot be monochromatic in the same color. The first system is the finite-sums setwhile singleton separated blocks in the second include every $x_i+2x_j+3x_k$ with $i<j<k$. Thus a sequence satisfying the proposed union would contradict that finite coloring.

Solved by gpt-5.6-sol high.