I'm currently reading through Jacod and Shiryaev (2002). From what I have understood so far, to show convergence in law of a sequence $(X^n)_{n \geq 0}$ of semimartingales we must show (i) that the sequence is tight so that its laws converge along a subsequence by Prokhorov's theorem and (ii) show that the limit is the same for any convergent subsequence.
In Theorem IX.3.21, assumption (iii) requires that there is a unique solution to the martingale problem, i.e., there is a unique probability measure under which the canonical process $X$ is a semimartingale with characteristics $(B, C, \nu)$, where those are the limits of the characteristics $(B^n, C^n, \nu^n)$ of $X^n$ in the right sense.
My question is: is this as an assumption about uniqueness only or is the assumption that $X$ is a semimartingale substantial?
Phrased differently: if a sequence $(X^n)_{n \geq 0}$ of semimartingales converges in law to $X$, is $X$ a semimartingale?