Odd-length linear equations are uncommon over the integers

Let $k\geq 3$ be odd, and let $a_1,\ldots,a_k\in\mathbb{Z}\setminus\{0\}$. Consider the equation

$$a_1x_1+\cdots+a_kx_k=0.$$

Call the equation uncommon over the integers when there is a 2-coloring of $[n]$ with fewer monochromatic solutions than a uniformly random 2-coloring.

Odd-length uncommonness conjecture. The equation is uncommon over the integers.

The claim appears in the paper's further remarks as a plausible general statement; no resolution is supplied in the provided text, so it remains open in this record.