Dimension-free comparison inequalities between $f$-divergences for Gaussian location mixtures

We establish dimension-free comparison inequalities between the Kullback-Leibler (KL) and ${\chi }^2$-divergences and the squared Hellinger distance for Gaussian location mixtures with compactly supported mixing distributions. We show that both divergences are bounded by the squared Hellinger distance up to constants depending only on the radius of the support, independently of the ambient dimension and the number of mixture components. These results resolve an open question raised by \cite{jia2023entropic} and strengthen existing comparison inequalities. Our proof relies on a Gaussian interpolation argument that relates the derivatives of these divergences to differences between posterior means.