The same one you'd have with any proof of ownership in a society with laws.
We trust online brokerages to maintain our ownership on securities with databases alone, but those are heavily regulated. The government isn't likely to enact regulation on baseball cards and action figures either.
So for safety we're pretty much down to paper documentation and a cumbersome sale process. That's slow and expensive, which is why rich people do it and regular people don't. Until now.
I don't see why people think blockchains have to exclusively do new and unprecedented things. It's productive to do old things more efficiently.