The monochromatic-solution lower-bound conjecture for linear equations

Let $[n]=\{1,2,\dots,n\}$, let $a_1,\dots,a_k\in\mathbb{Z}$, and consider the equation

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

For a coloring $f:[n]\to\{-1,1\}$, call a solution $(x_1,\dots,x_k)\in[n]^k$ monochromatic if $f(x_1)=\cdots=f(x_k)$. Monochromatic-solution lower-bound conjecture. Every coloring has $\Omega(n^{k-1})$ monochromatic solutions over $[n]$. The conjecture asserts a positive-order proportion of the $C n^{k-1}+O(n^{k-2})$ total solutions, but its status is not resolved in the supplied text.