The prime power conjecture for linear perfect codes in $A_n$

Let $A_n$ be the root lattice equipped with the metric $d$, and let $d_a$ denote the corresponding metric on $\mathbb{Z}^n$. A linear $1$-perfect code is a sublattice whose radius-$1$ balls are disjoint and cover the ambient metric space.

Prime power conjecture. There exists a linear $1$-perfect code in $(A_n,d)$, equivalently in $(\mathbb{Z}^n,d_a)$, if and only if $n$ is a prime power.

Existence when $n$ is a prime power follows from the corresponding difference sets, while necessity is open. The claim is equivalent to the prime power conjecture for Abelian planar difference sets.