For two random variables $X$ and $Y$ taking values in $\mathbb{R}^m$, the convex distance $d_c$ is defined as

$$d_c(X,Y) = \sup_{h} \lvert \operatorname{E}(h(X)) - \operatorname{E}(h(Y)) \rvert,$$

where the supremum is taken over all indicator functions of measurable convex subsets of $\mathbb{R}^m$.

For $m=1$, it is easy to see that $d_c$ coincides with the Kolmogorov distance whenever $X$ and $Y$ are continuous, i.e. we have that $d_c(X,Y)= \sup_{x \in \mathbb{R}} \lvert F_X(x) -F_Y(x) \rvert$, where $F_X$ and $F_Y$ denote the cumulative distribution functions of $X$ and $Y$, respectively.

In particular, if $m=1$, we have that if a sequence $(X_n)$ of continuous real-valued random variables converges to another random variable $Y$ in distribution, then $d_c(X_n,Y) \to 0$ as $n \to \infty$. Does this implication continue to hold if $m \geq 2$?

I could neither find a reference, nor a proof of this myself.