The Golomb--Welch strong conjecture for perfect Lee codes

For integers $n,e$, let a $PL(n,e)$-code mean a perfect $e$-error-correcting code in the Lee metric on $\mathbb{Z}^n$. Golomb--Welch strong conjecture. There is no $PL(n,e)$-code for $n\geq 3$ and $e\geq 2$. This is the Euclidean-space, or unrestricted-periodicity, strengthening of the weak formulation. The paper proves many parameter ranges, including sufficiently large $e$ for each fixed $n$, but states that the conjecture remains open.