Queue-Number of Planar Graphs

Does every planar graph have $O(1)$ queue-number? A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into non-nested queues. Edge $xy$ is nested inside edge $vw$ if $v<x<y<w$ in the linear order. The queue-number of a graph $G$ is the minimum number of queues in a queue layout of $G$. This question amounts to asking whether every planar graph has a vertex ordering with a constant number of pairwise nested edges (called a rainbow).