Free boundary minimal surfaces with the same symmetry group as one of the Platonic solids

Conjecturally, the Euclidean unit ball contains families of free boundary minimal surfaces with increasing genus and the same symmetry group as one of the Platonic solids. Some of these conjectured surfaces come in pairs with the same topology and symmetry group. The free boundary minimal surfaces of genus zero with tetrahedral, octahedral or icosahedral symmetry could be included as first members of these families.

Free boundary minimal surfaces with tetrahedral symmetry

Free boundary minimal surfaces with octahedral symmetry

Free boundary minimal surfaces with icosahedral symmetry