Hmmm... but if I put tubes in, then I won't be able to use convenient plugs to patch them, right? Then again, since they're not holding air particularly well in the first place, possibly no great loss. Still, I don't relish the thought of having to re-seat the tires every time I need to patch the tube. As for foam, I vaguely recall reading that my warranty is void if I put foam in my tires, so that's out.
It's not old rubber, as the tractor itself is pretty new, even if you count sitting-on-the-lot-time. My current best guess is a slow leak that didn't show up on the under-water bubble test.