Before I gave up, I would re-name your troublesome folder. Then re-create a new save folder from within the program, then reload the affected project from the re-named old folder and save it to the new folder, or just copy the old folder contents if that works.
Check that effective permissions is the same for all the troublesome files not just the folder, as there can be a discordance between file and folder permissions.
Perhaps there is a stuck leftover inheritance issue? Nominally inherited permissions should show as grayed out users in the list, but your system is not behaving nominally. Try disabling permissions inheritance in the troublesome folder. Alternatively go to the top level folder (parent) for your troublesome folder, check that it has the required permissions and then try to reset the child folders to that permissions set by choosing to "Replace all child object permissions with inheritable permissions from this object." in the parent folder to see if that will reset the permissions in your troublesome folder.
Or stand on one leg and scream like a chicken.