That's talking about being liable for damages. The only thing that's asserted under penalty of perjury is that you're authorized to act on behalf of a copyrigh

That's talking about being liable for damages. The only thing that's asserted under penalty of perjury is that you're authorized to act on behalf of a copyright holder. If you lie about everything except that one fact, the only thing that can happen to you is someone can try (and good luck with that...) to claim damages from you. The only way you face perjury charges is if you lie about being authorized to act for the copyright holder.