We prove that the set of the smallest eigenvalues attained by $3$-colorable graphs is dense in $(-\infty, -λ^*)$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$ and $ρ$ is the positive real root of $x^3 = x + 1$. As a consequence, in the context of spherical two-distance sets, our result precludes any further refinement of the forbidden-subgraph method through the chromatic number of signed graphs.