Determine commonness of odd-term equations for two colors

Determine whether every linear equation with an odd number of terms greater than three is 2-common or 2-uncommon over the integers.

Background

Prior work cited in the paper established that all 3-term linear equations are 2-uncommon over the integers and proposed a characterization of 2-common equations. The paper notes that this characterization was later disproved by a 4-term example, while the odd-term case beyond three terms remained unresolved at the time of writing. The present paper resolves only the corresponding r-commonness question for r at least 3, not the two-color case.