A Skorohod representation theorem for uniform distance

Let $\mu_n$ be a probability measure on the Borel $\sigma$-field on $D[0,1]$ with respect to Skorohod distance, $n\geq 0$. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are $D[0,1]$-valued random variables $X_n$ such that $X_n\sim\mu_n$ for all $n\geq 0$ and $\norm{X_n-X_0}\rightarrow 0$ in probability, where $\norm{\cdot}$ is the sup-norm. Such conditions do not require $\mu_0$ separable under $\norm{\cdot}$. Applications to exchangeable empirical processes and to pure jump processes are given as well.