Schwarz $\mathrm{P}$ and Schwarz $\mathrm{D}$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas

Let $T$ be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides $2,\sqrt3,\sqrt3$, so that the dihedral angles are $π/2$ along one opposite pair of edges and $π/3$ along the remaining four. By Smyth's theorem $T$ contains three embedded minimal disks meeting $\part