Lightning isn't as private as commonly thought. Lightning has good sender privacy, but not so great recipient privacy. I think it will take a combination of upgrades to improve this.

An article with more info and also lists some privacy improvements at the end: https://abytesjourney.com/lightning-privacy/

Audio reading of the article: https://www.listennotes.com/podcasts/bitcoin-audible/read_616-current-state-of-4hXEJBxez79/