Periodic striped ground-state conjecture for the two-dimensional dipolar Ising model

Consider the two-dimensional Ising model on the periodic torus

$$\Lambda_L=\mathbb Z^2/L\mathbb Z^2$$

with nearest-neighbor ferromagnetic coupling $J$, Hamiltonian $\mathcal H_L$, periodic striped configurations $\bm{\sigma}_s(h)$ of width $h$, and stripe energy per site $\mathcal E_s(h)=\mathcal E(\bm{\sigma}_s(h))$. Let $h^*=h^*(J)$ minimize $\mathcal E_s(h)$ over $\mathbb N$, and let $e_0$ be the thermodynamic-limit ground-state energy per site. Periodic striped ground-state conjecture. There exists $J_0>0$ such that, for every $J\geq J_0$ and every integer $L$ divisible by $2h^*$, the minimizers of $\mathcal H_L$ are precisely $\bm{\sigma}_s(h^*)$, its translations, and its discrete rotations. In particular,

$$e_0=\mathcal E_s(h^*).$$

The stated result includes the exceptional-coupling case described in the source: when the stripe energy has two contiguous minimizers and $L$ is divisible by $2h^*(h^*+1)$, the corresponding stripes of width $h^*+1$ and their translations and rotations are additional minimizers. The conjecture concerns the characterization of ground states in the large-$J$ regime; the source explicitly states that it remains open.