The recipe calls for oak and cherry, I think, but the important thing is it calls for a lot of smokewood-- relative to other recipes-- but you are directed to burn it down quite a bit before putting the meat on. And that's a good idea just from a practical standpoint, as it would be way too much smoke otherwise.
I'd vote for hickory, and very little mesquite or leave it out altogether. Those are both strong woods, and together they may overpower things. Save the mesquite for a brisket.