is the limit of a Markov process sequence a Markov process?

Question

Assume $\{ X_t^n, \: t \geq 0 \}$ is a sequence of Markov processes (think in terms of diffusions) defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that there exists $\{ X_t, \: t \geq 0 \}$ a process defined on the same probability space such that $$ \lim_{n \to \infty} \mathbb{E} \left ( \sup_{0 \leq t \leq T} |X_t^n - X_t |\right ) = 0. $$ Is it clear that X is a Markov process?