Suppose we are given a class of probability density functions $\mathcal{F}$ so that for every $f \in \mathcal{F}$ we have $\alpha \leq f \leq \beta$ for some positive $\alpha, \beta \in \mathbb{R}_+$ and the density have the same support set $B$.

Here it is proved that in this setting the $L_2$ norm between density functions, i.e., $$ \|f-g\|^2_2 = \int_B (f(x) - g(x))^2 \mu(dx) $$ is equivalent to the KL divergence between $f$ and $g$, i.e., for $$ D_{\operatorname{KL}}(f \| g) = \int_B f(x) \log \frac{f(x)}{g(x)} \mu(dx) $$ we have $$ D_{\operatorname{KL}}(f \| g) \asymp \|f-g\|^2_2 $$ where $\asymp$ denotes equiality up to constant factors (which may depend on $\alpha, \beta$ but we consider those fixed).

Here a question was asked regarding the comparison of $L_1$ and $L_2$, and was argued that for such $\mathcal{F}$ classes of functions, it is possible that $$ \|f-g\|_1 = \int_B |f(x) - g(x)| \mu(dx) \ll \|f-g\|_2 $$ in general.

By Pinsker's inequality we know that $$ \|f-g\|_1 \lesssim \sqrt{D_{\operatorname{KL}}(f \| g)} \asymp \|f-g\|_2 $$ where $\lesssim$ indicates inequality up to absolute constant factors.

My question is as follows. Suppose we impose smoothness assumptions on $\mathcal{F}$. For instance suppose $\mathcal{F}$ consists of Lipschitz continuous functions with Lipschitz constant $L$. Does a reverse Pinsker's inequality hold in this case i.e. is it possible that $$ \|f-g\|_1 \gtrsim \sqrt{D_{\operatorname{KL}}(f \| g)} \asymp \|f-g\|_2\;? $$

The counterexample given here fails to work in the Lipschtiz $\mathcal{F}$ case, and is not clear how it can be recreated because the functions need to be smooth.