Suppose we have two Gibbs measures with densities $$ p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)). $$ Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, that is, $$ D(f, g) := \text{KL}(p_f \| q_g). $$

Question: Do we have the following lower bound: $$ D(f, g) \geq \|f - g\|^2, $$ where we are interested in, for example, the $L_2$-norm of $f-g$.