On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint

Kosek, Robertson, Sabo, and Schaal studied the minimum number $M_k(n)$ of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in $2$-colorings of $[k+1,k+n]$. They proved that for every fixed $k\ge 0$, $M_k(n)=\frac{n^3}{12(1+2\sqrt{2}{)}^2}(1+o_k(1)),$ and left open the case $k\le -2$. In this paper, we resolve that remaining range.