While it looked for a while like director Adam Wingard was going to become Legendary's in-house Godzilla and Kong director, it looks like the pair are set for a bit of a break. The Hollywood Reporter has said that Wingard will not be directing the next Godzilla and Kong film, and that the reason is nothing to really be worried about.
The report notes that the issue mainly stems from timing problems as Wingard intends to work on a different project before helming another MonsterVerse epic. This is despite the fact that he has stated multiple times that he has ideas and plans for more stories in the action-packed world.
Legendary is leaving the door open for Wingard to return and helm another project in the future but it does seem like a different person will be in the director's chair for the next MonsterVerse clash.