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The treaty which governs this is the Montreal Convention.
Article 36 and chapter 5 seem to indicate that you can claim from either, though I’m a bit too tired at this time to sift through the legalese.
I would personally claim from United since they are the ones you contracted with.
Don’t forget to mention “article 19 of the Montreal Convention” in your claim. Note the limits on responsibility (about $1700 per passenger).
See also https://www.forbes.com/advisor/credit-cards/travel-rewards/montreal-convention-luggage/ which explains a lot of it in much more accessible language and gives quite a few important details.