The cyclicity conjecture for Abelian planar difference sets

Let $A_n$ be the root lattice, let $d$ be its metric, and let $\mathcal L$ be a $1$-perfect code in $(A_n,d)$. For $(i,j)\in\{0,1,\ldots,n\}^2$, let $\mathbf f_{i,j}$ denote the corresponding direction, and define the period of $\mathcal L$ along that direction as the least positive translation length preserving $\mathcal L$.

Cyclicity conjecture for Abelian planar difference sets. The period of $\mathcal L$ in $A_n$ along $\mathbf f_{i,j}$ is equal to $n^2+n+1$ for at least one vector $\mathbf f_{i,j}$, with $(i,j)\in\{0,1,\ldots,n\}^2$.

This is the code-theoretic form of the conjecture that every Abelian planar difference set lies in a cyclic group. It is presented as an important unsolved conjecture; the equivalence uses the isomorphism between the ambient quotient $A_n/\mathcal L$ and the group containing the difference set.