Nonexistence conjecture for perfect codes over non-prime-power alphabets

Let $q \ge 6$ be a non-prime power, and let $Z_q$ denote the alphabet of size $q$.

Nonexistence conjecture. Let $q \ge 6$ be a non-prime power. Then, there are no perfect error correcting codes over $Z_q$.

Existing results establish nonexistence for several values of $q$ and error-correction parameters, but the general classification over non-prime-power alphabets remains open.