It's simple to prove that the theorem works 100% in all cases where countries are shaped as a square of equal size (like a chessboard), but if a map has a country whose horizontal length is longer by twice or more compared to other countries then the theorem succeeds if the length is longer by a multiple of odd numbers, fails if longer by a multiple of even numbers.
That's only if a common point is counted as a borderline. If a common point does not get counted as such, the success rate is no more than 3 out of all infinite cases.
If the square-shaped countries were situated only along one single diagonal line in an imagined world, so that their only point of contact with each other was a common point, then only one color may be used to color them all. If so, the Prof's student would not even have bothered to ask the original question.
So far there has not been a single point raised for any errors or omissions or faulty or incomplete logic in the way this conclusion was reached.
To wit, if the conclusion were correct, it would be too obvious not to have been noticed and looked at and discarded or accepted by everyone who ever gave it a thought in the last 100 years or so.