Schützenberger's maximal-code commutative equivalence conjecture

Let $C$ be a finite uniquely decodable code over the binary alphabet $\{a,b\}$. A code is maximal if, for every word $w\in\{a,b\}^{<\mathbb{N}}\setminus C$, the set $C\cup\{w\}$ is not a code. Two codes are commutatively equivalent if there is a bijection between them preserving the numbers of $a$'s and $b$'s in corresponding words. A code is prefix-free if no codeword is a proper prefix of another.

Schützenberger's conjecture. Every maximal code is commutatively equivalent to a prefix-free code.

The conjecture is presented as a restricted version of the disproved unrestricted conjecture. The supplied text does not provide a resolution of this maximal-code version; its status is therefore left open.