I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :

$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \mathbb{E}[X_1 ^k] ) ^2$ where $X_1$ and $X_2$ are random variables whose laws are $\mu_1$ and $\mu_2$ respectively.

Let $d_{TV}$ be the total variation distance on the same space. I'm considering distributions lying in a set $\Sigma$ with :

$\Sigma = \{ \alpha U(s,t) + (1-\alpha)W_1(R) \lvert (s,t,R,\alpha) \in [0,\epsilon_1]^2 \times [0,\epsilon_2] \times [0,1] \} $

where :

$ \epsilon_1, \epsilon_2 >0 $, $U(s,t)$ is the uniform distribution on $[s,t]$ and $W_1(R)$ is the semi-circular law of parameter $R>0$ and centered on 1.

I want to know if there exists a constant $C>0$ (possibly for $p$ big enough) such that

$\forall (\mu_1,\mu_2) \in \Sigma^2, \ d_{TV}(\mu_1,\mu_2) \leqslant C d_p(\mu_1,\mu_2) $.

Is is possible to possible to find such a constant when $\mu_1$ and $\mu_2$ are both uniform distributions for instance but I do not know if it's true in the general case. Do you have an idea or reference for this problem? Any suggestion is welcome!

Thanks.