Compositions of maximal codes

The proposition “For composable thin codes Y and Z, the composition Y∘Z is maximal if and only if Y and Z are maximal” put forward by J. Berstel and D. Perrin in their book “Theory of Codes” is well known. Is the proposition also true without the assumption that Y and Z are thin? We give an example showing that the answer is negative. Furthermore, several generalizations of the above proposition are also given.