Some claim it's probably very real (and very soon):

https://comicbook.com/gaming/news/the-last-of-us-part-3-release-date-report/

In June 2023, game series creative director and HBO show cocreator Neil Druckmann mentioned they had other priorities, though:
https://www.polygon.com/23572852/the-last-of-us-part-3-release-naughty-dog-announcement-story

Yet, there are clues to a "yes" everywhere – but not necessarily for 2024:

https://thedirect.com/article/the-last-of-us-part-3-release-story-details

And at the end of the day, TLOU II unnecessary remake may not perform (financially) that well, thus increasing the odds of having a part 3 of an already successful franchise, much like the way it ended up being with Naughty Dog's Uncharted, with 4 installments, a sequel game, and a Hollywood movie.