Let $\mathcal P(\Omega)$ be the set of probability measures supported on some compact subset $\Omega\subset\mathbb R^d$. For $\mu\in\mathcal P(\Omega)$, denote by $F_{\mu}$ its characteristic function, i.e.

$$F_{\mu}(x)~:=~\int_{\mathbb R^d}e^{i\langle x,y\rangle}\mu(dy)~=~\int_{\Omega}e^{i\langle x,y\rangle}\mu(dy),\quad \forall x\in\mathbb R^d.$$

Let $W(\cdot,\cdot)$ be the Wasserstein distance of order $1$. My question is whether there exists a continuous function $c:\mathbb R_+\to\mathbb R_+$ with $c(0)=0$ s.t. it holds for all $\mu$, $\nu\in\mathcal P(\Omega)$:

$$W(\mu,\nu)~\le~c\big(\|F_{\mu}-F_{\nu}\|\big),\quad \mbox{with } \|F_{\mu}-F_{\nu}\|:=\max_{x\in\mathbb R^d}|F_{\mu}(x)-F_{\nu}(x)|.$$

Any answer, comments and references are highly appreciated!