Let $k$ be a perfect field of characteristic $p>0$. Let $K$ be a finite, totally ramified extension of $K_0:=\mathrm{Frac}\ W(k)$ and let $\mathcal{O}_K$ be the ring of integers of $K$. All group schemes in this question will be assumed to be commutative. If $\mathscr{V}$ is a finite flat group scheme over $\mathcal{O}_K$, we say that $\mathscr{V}$ is $\mathit{maximal}$ if for any other finite flat (commutative) group scheme $\mathscr{W}$ over $\mathcal{O}_K$ which has the same generic fibre as $\mathscr{V}$, there is a morphism $\mathscr{V}\to\mathscr{W}$ which restricts to the identity on the generic fibre. By a theorem of Raynaud, we can replace any finite flat group scheme over $\mathcal{O}_K$ by a maximal one without changing the generic fibre.

Now the question: Let $\mathscr{V}$ be a finite flat group scheme as before and suppose that $\mathscr{V}$ is maximal. Is it true that the connected component $\mathscr{V}^0$ of $\mathscr{V}$ is maximal as well?