Thanks everyone.
It looks like some of you are attaching the rack to one wall (as I'd prefer to do), and others are spanning both walls. I plan on installing cabinets up front, so I'm reserving space up there for that.

Another idea was suggested to me that I add a back-brace to my tire rack and screw through that so I can hit each wall stud along the length of the rack. I might give that a try.

Thanks again.
John