Global minimality conjecture for the magnetic Ginzburg–Landau solutions

Let $(A,\psi)$ denote the entire magnetic Ginzburg–Landau solutions constructed in Theorems 1.1 and 1.2, and let the corresponding energy functional be the magnetic Ginzburg–Landau energy.

Global minimality conjecture. The solutions of the magnetic Ginzburg–Landau equations provided in Theorem 1.1 and Theorem 1.2 are energy minimizers of the energy functional.

This conjecture is motivated by the analogous conjecture for Allen–Cahn solutions associated with the area-minimizing Simons cone. In the present four-dimensional setting, the union of the two planes is also area minimizing, but the source does not provide a resolution of the magnetic Ginzburg–Landau minimality claim.