Four-direction LP asymptotics

Let $\alpha$ be the middle real root of

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

For odd $n=2m+1$, let $L_m^{\mathrm{fat}}$ and $L_m^{\mathrm{thin}}$ denote the relaxation optima on the fat and thin colour classes, respectively. For even $n=2m$, let $L_m^{\mathrm{even}}$ denote the relaxation optimum on either colour class. Four-direction LP asymptotics. The three symmetry types have the common limiting ratio

$$\lim_{m\to\infty}\frac{L_m^{\mathrm{fat}}}{2m+1} = \lim_{m\to\infty}\frac{L_m^{\mathrm{thin}}}{2m+1} = \lim_{m\to\infty}\frac{L_m^{\mathrm{even}}}{2m} = \alpha.$$

The conjecture is motivated by computed reduced linear programs in all three symmetry types and by an exact continuum upper-bound certificate in the odd-fat case. The computations suggest a common limiting slope, but do not establish the asymptotics for the finite LPs.