Divisibility conjecture for the index of self-dual ternary codes

Let $C$ be a self-dual ternary code of length $c$. Its index is the number of even maximal codewords minus the number of odd maximal codewords. Index divisibility conjecture. The index of any self-dual ternary code is divisible by $24$, even if the length is merely divisible by $12$. For every length $c=12k$, there exists a self-dual ternary code of index $24$. The preceding theorem proves divisibility by $24$ when the length is divisible by $24$, and by $12$ when it is merely divisible by $12$; the stronger divisibility and existence assertions remain open.