Saad–Wolf commonness conjecture for even-variable linear equations

Let $q$ be a prime power, let $n$ be a positive integer, and let $k\ge 2$. Consider the linear equation over $\mathbb{F}_q^n$

$$\na_1x_1+\cdots+a_{2k}x_{2k}=0.$$

The equation is common if, for every two-coloring of $\mathbb{F}_q^n$, the number of monochromatic solutions is at least $2^{1-2k}q^{n(2k-1)}$. Saad–Wolf's conjecture. The equation is common in $\mathbb{F}_q^n$ if and only if the multiset of coefficients $\{a_1,\ldots,a_{2k}\}$ can be partitioned into $k$ pairs, each pair summing to zero. Saad and Wolf proposed this characterization for linear equations with an even number of variables; the paper's abstract states that the authors characterize common linear equations and that their construction disproves some such properties, but the supplied material does not establish the resolution of this exact conjecture.