Yes, I agree. Maybe just reiterate what OP said in private. Without making a big scene, saying something like "Hey, you're doing that thing again where you're making it seem like you're one upping me, and you're making me feel diminished." There's really nothing to say to that, depending on their reaction, it'll show you whether to cut this person off or not

Talk to them privately, and ask why they're doing that. I forgot where, but I've read that sometimes people do unconsciously do that because they want to connect by sharing a similar experience. They may not notice what they're doing and not realize how that makes you feel.

If they still do it after, call them out in public there and then. You already did your part by being thoughtful and speaking in private. No need to keep that up if they still do it.