Etzion's conjecture on diameter perfect Lee codes

Let $DPL(n,d)$ denote a diameter-$d$ perfect Lee code of word length $n$.

Etzion's conjecture. There is no $DPL(n,d)$ code for $n\geq 3$ and $d>4$, with the exception of the pair $(n,d)=(3,6)$.

The conjecture extends the Golomb–Welch conjecture to diameter perfect Lee codes. The exceptional $DPL(3,6)$ code exists by a Minkowski tiling, while the general nonexistence assertion remains open.