Yes, unfortunately they can. As long as the board votes 2/3 to go into executive session, the board can discuss the character and competence of an employee in an executive session. They can’t, however, fire the employee in that session: they have to emerge from the session and take a public vote in accordance with the Open Meetings Law. Notably, the employee has the right, under La. R.S. 42:17(A)(1), to require that the discussion of their character and competence be held in public.