Does $\mathbb{E}(|X_t|) = O(t)$ hold for a Lévy process $(X_t)_{t \geq 0}$?

Question

For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does $\mathbb{E}[|X_t|]=O(t)$ hold? Assume there is no continuous part, and that the process has bounded jumps (the absolute moment of the compound Poisson part is O(t)), so: \begin{align*} X_t = \lim_{\epsilon\to 0}\int_{\epsilon<x<1,s\leq t}x\bar J(ds,dx) \end{align*} where $\bar J(ds,dx)=J(ds,dx)-\nu(dx)ds$ is the compensated jump measure.