3 problems

Optimality conjecture for triangular and square lattices. There exist $1<p_1<p_2$ and $1<p_3<p_4$ such that, up to rotation, the following hold:

Triangular-lattice uniqueness conjecture. The triangular lattice $\Lambda_{A_0}$ is the unique solution of $(P)$.

Large-area square-lattice conjecture. If $A$ is sufficiently large, the square lattice is the unique solution of $(P_A)$.