Mine didn’t with both Avon and Metzler. The bike had to be jacked up high enough for the tire to come straight out and then tilt at an angle and slide under the can.
I will try that next time now that I figured out how to add a strap for stability on that lift. I do have the larger diameter Bridgestone Exedra Max on there (and replaced with same) which also has more of a lip where the tread meets the sidewall.