I have seen a proof for the following statement: If a counting process $\left\{ N(t) \mid t\ge 0\right\}$ is homogeneous and has independent increments, i.e., $N(b_1)-N(a_1)$, $\dots$, $N(b_n)-N(a_n)$ are independent whenever $(a_1, b_1]$, $\dots$, $(a_n, b_n] \subseteq \mathbb{R}_{\ge 0}$ are disjoint, then it is a Poisson process.
But I wonder if the statement remains true also when the process is inhomogeneous. I.e., is any counting process with independent increments a Poisson process?