Monochromatic no-three-in-line asymptotics

Let $D_{\mathrm{mono}}(n)$ denote the exact maximum number of points in a monochromatic no-three-in-line configuration on the checkerboard grid of side length $n$, and let $\alpha$ be the middle real root of

$$401\alpha^3-1744\alpha^2+2240\alpha-768=0.$$

Monochromatic NTIL asymptotics. The exact monochromatic no-three-in-line maxima satisfy

$$\lim_{n\to\infty}\frac{D_{\mathrm{mono}}(n)}{n}=\alpha.$$

This is stronger than the corresponding relaxation asymptotics: the finite LP computations motivate it, while the continuum certificate gives an exact upper bound only for the odd-fat relaxation. The discrete limit remains unproved.