Conjecture on linear perfect codes in the $l_2$ metric

Let $n$ be a positive integer, let $r>0$, and let $\Lambda$ be an additive subgroup of $\mathbb{Z}^n$. A linear perfect code in the $\ell_p$ metric is such a subgroup whose translates tile $\mathbb{Z}^n$ by the balls

$$B_p^n(r)=\left\{(x_1,\ldots,x_n)\in\mathbb{Z}^n:|x_1|^p+\cdots+|x_n|^p\leq r^p\right\}.$$

Conjecture. There are no linear perfect codes with parameters $(n,r,2)$ except when $r=1$ or

$$(n,r)\in\left\{(2,\sqrt{2}),(2,2),(2,2\sqrt{2}),(3,\sqrt{3})\right\}.$$

In particular, there are no linear perfect codes in the $\ell_2$ metric if $n>3$ and $r>1$. The claim is motivated by computational evidence and by the absence of cubic polyominoes in the relevant higher-dimensional cases; it remains open in the source.