Convergence of $\mathbb{E}|\int X(\omega,x) (m_k(dx)-m(dx))|^2$ when $m_k(x)dx$ tends to $mdx$ in Wasserstein 2 distance

Question

In $\mathbb{R}^n$, we suppose that $m_k$, $m$ are probability densities with finite second moment such that $m_k(x)dx$ tends to $mdx$ in Wasserstein 2 distance as $k \to \infty$. If $X(\omega,x)$ is a random variable such that $\mathbb{E}(|X(\omega,x)|^2) <1$ for any $x$, can I conclude that $\mathbb{E}|\int X(\omega,x) (m_k(x)dx-m(x)dx)|^2$ tends to $0$ as $k \to \infty$?