Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.

We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective decomposition $D\sim D_1+D_2$ we have $D_1\cdot D_2\geq1$. Notice that these two notions are not equivalent (if $D$ is not reduced), e.g. $D=2E$ where $E$ is an elliptic curve, is numerically disconnected.

Assume $h^1(D)=0$. The ideal sheaf sequence of $D$ yields $h^0(\mathcal{O}_D)=h^0(\mathcal{O}_X)=1$ in cohomology. Hence $D$ is connected. My question is:

Does $h^1(D)=0$ imply numerical connectedness?