No, you need to read the actual law, 17 USC 513(c)(3)(A)(3), and the cited cases.
The statute on its face doesn't require actual facts of infringement to be included in the notice, only a good faith assertion that infringement (or use in infringing activities) is occurring. Consequently, that allegation is sufficient for purposes of the DMCA notice. Indeed, that is the holding of the cited cases, which are included with the notice to show that courts have found that a good faith belief of infringement is sufficient.
The statute on its face doesn't require actual facts of infringement to be included in the notice, only a good faith assertion that infringement (or use in infringing activities) is occurring. Consequently, that allegation is sufficient for purposes of the DMCA notice. Indeed, that is the holding of the cited cases, which are included with the notice to show that courts have found that a good faith belief of infringement is sufficient.