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And the law makes no mention that the branded product has to be provided for free.
Actually it does. I'll highlight it.
"(c) Prohibition on conditions for written or implied warranty; waiver by Commission
No warrantor of a consumer product may condition his written or implied warranty of such product on the consumer's using, in connection with such product, any article or service
(other than article or service provided without charge under the terms of the warranty) which is identified by brand, trade, or corporate name; except that the prohibition of this subsection may be waived by the Commission if—
(1) the warrantor satisfies the Commission that the warranted product will function properly only if the article or service so identified is used in connection with the warranted product, and
(2) the Commission finds that such a waiver is in the public interest."
I know in the past that was talk about whether Oil Certs fall under the concept of "brand" as a sort of grey area. This is why automakers may deny engine warranties when an engine is full of sludge because it's obvious that the oil was no longer performing regardless of the brand used. They can't deny a claim under factory warranty when an OE equivalent filter failed.