I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we cannot upper bound the total variation by Wasserstein distance. Many answers in MO give the same intuitive counterexample: consider a discrete distribution and a continuous distribution.

However, assuming the two underlying distributions are both continuous (for example consider the simpliest when they are both Gaussian mixtures with zeros mean or just Gaussians with zero mean), can we find such an upper bound?

Any related suggestions or comments are welcomed!