I did some shopping and research today, posting the results in case it helps anyone. I built a mockup of the size of a folded Stern from PVC pipe, and used it to test several vehicles at a few dealerships.
The test piece is 30.75 inches in height, very slightly oversized for a Stern which is about 30.25, and a Bally/Williams is slightly shorter at about 29.5.
Honda: 2021 model year HR-V and CR-V both fit very well - both of these have the fully flat folded seat backs, that the 2023 models don't.
2023 HR-V and CR-V are both right on the borderline - the mockup fit with some pushing but the pipes were bent slightly so it's hard to tell for sure. The 2023 CR-V in particular has an awkward lump in the folded back seat that is troublesome. You could probably do it with a trick like sliding the backbox towards the coin door or loading the machine on its side.
Hyundai: Definitely no for the 2023 Venue and Kona. Yes for the Tucson, fully fit easily.
Toyota: No for the 2023 Corolla Cross and Venza. Borderline yes for the Rav-4 but with millimeters to spare for clearance.
Chevrolet: No for the 2023 Trax and Trailblazer. Borderline but probably no for the Equinox and Blazer, the hatch clearance on each is about an inch too short, maybe yes sideways or by sliding the backbox.
I didn't get to Kia today but intend to try a Seltos and Soul next week.
(Of course, take anyone else's firsthand information as more authoritative than my mockup.)