Before revision:  We show that for any k \geq 0, the class of B_k-CPG graphs is strictly contained in the class of B_{k+1}-CPG graphs even within the class of planar graphs, thus implying that there exists no k \geq 0 such that every planar CPG graph is B_k-CPG.
After revision:  We first show that, for any k \geq 0, the class of B_k-CPG graphs is strictly contained in the class of B_{k+1}-CPG graphs even within the class of planar graphs, thus implying that there exists no k \geq 0 such that every planar CPG graph is B_k-CPG.