Golomb–Welch conjecture on perfect Lee codes

Let $\mathcal A_q^n$ be the ambient Lee-metric space, and let a perfect $(n,M,d)_q$-Lee code be a Lee code with $M$ codewords whose packing and covering radii coincide. Golomb–Welch conjecture. If $d$ is odd, then there exist no perfect $(n,M,d)_q$-Lee codes for

$$n\geq 3,\qquad q\geq d\geq 5.$$

The conjecture concerns the existence of perfect tilings by Lee-metric balls and has been studied since Golomb and Welch's formulation. The source gives no resolution, so the conjecture remains open.