In several recent works on infinite-dimensional ODE systems 6,7,9,11 arising from mean-field limits of stochastic agent-based models in economics and social sciences, the Gini index often emerges as a natural Lyapunov functional along solutions. It is also shown to converge to the Gini index of the equilibrium distribution. However, it remains unclear whether such convergence implies convergence of the full probability distributions or stronger forms. In this paper, we address this question by proving several results that clarify the relationship between the Gini index and other metrics for distributional closeness, such as the Wasserstein and ? ??? distances.