You must claim _under penalty of perjury_ that some work is infringed and you are or represent the copyright holder.
False. Only the second part is required to be under penalty of perjury: the part where you claim that you are or represent the copyright holder of the work alleged to be infringed; the allegation of infringement is not under a penalty of perjury, even if it's obviously bogus.
If you do commit this perjury; it is not going to be provable by a third party, unless the copyright work you are alleging to be infringed is someone else's work, and the third party can prove you don't represent the copyright holder.
If the DMCA letter writer slaps on an allegation of infringement of their unpublished title Xyz; nobody can prove it's perjury, at least, without the admission of the person who sent the letter.
See the text from the act:
A statement that the information in the notification is accurate, and under penalty of perjury, that the complaining party is authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.