Horak–AlBdaiwi conjecture on diameter perfect Lee codes

Let $DPL(n,d)$ denote a diameter $d$ perfect Lee code in $\mathbb{Z}^{n}$. For odd $d$, this agrees with a perfect Lee code of radius $(d-1)/2$; for even $d$, it is defined using a tiling by maximum-size anticodes of diameter $d-1$. Horak–AlBdaiwi conjecture. There is no $DPL(n,d)$ code for $n\ge3$ and $d>4$, with the exception of the pair

$$(n,d)=(3,6).$$

This is presented as an extension of the Golomb–Welch conjecture. Etzion proposed the problem in 2011, and Horak and AlBdaiwi formally presented the conjecture in 2012; the source does not state whether it has been resolved.