It is uncertain whether Neymar will stay with PSG. However, PSG management is not interested in keeping Neymar in the squad. Neymar was a completely losing project for PSG. Neymar was signed by PSG for a record transfer fee. But Neymar failed to live up to PSG's expectations.
With Messi leaving the team, PSG may now try to sell Mbappe as well. Because PSG will be able to make a lot of money by selling Mbappe now. I would say PSG management should sell Mbappe, Neymar, and Messi all and form a completely new squad. And they should prioritize young players. And before that, PSG should look for an experienced coach.
Losing of Messi and Neymar were never been issue for the PSG but now suddenly losing of Mbappé is going to be terrible for them even now I have feeling most chances if Mbappé decided to leave then management can sell this club as well because this will lose worth and surely need to be good amount for setting as par current standard, and they already announce few months back if they fail to fill their dreams they will sell this club so now most chances things will be gone in this way, and we will have completely new-look in next season which will be good for the French League because mostly teams will be on same level and competitions could be tough for the top spots as well.
PSG management will never allow Mbappé to leave this club as free agent so most chances these all things will be cleared in next few weeks with most chances in next season we will have PSG with all new faces and completely different side from this old era.