Golowich's conjecture on regularity of large homogeneous equations

Let a linear homogeneous equation have nonzero integer coefficients, not all of the same sign. Golowich's conjecture. For each positive integer $k$ there is an integer $m(k)$ such that for any $m\ge m(k)$, any such equation in $m$ variables is $k$-regular.

This conjecture predicts that sufficiently many variables force bounded color-regularity for every fixed number of colors. The source states that it is false, so the proposed uniform threshold $m(k)$ does not exist in the asserted generality.