Acyclic edge-coloring of planar graphs: $\Delta$ colors suffice when $\Delta$ is large

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $\chi'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $\chi'_a(G)\ge \Delta(G)$ for every graph $G$. Cohen, Havet, and M\"{u}ller conjectured that there exists a constant $M$ such that every planar graph with $\Delta(G)\ge M$ has $\chi'_a(G)=\Delta(G)$. We prove this conjecture.
Comment: 15 pages, 8 figures; minor revisions to incorporate reviewer feedback; to appear in SIAM J. Discrete Math