Free boundary minimal surfaces of genus two

Conjecturally, the Euclidean unit ball contains a family of free boundary minimal surfaces with genus 2, any number of boundary components and either pyramidal symmetry group of order 6 or prismatic symmetry group of order 12. Curiously, the topology does not necessarily determine a surface in this family uniquely, as illustrated by the examples with four and five boundary components. A free boundary minimal surface with genus 2 and connected boundary and two more conjectural examples with genus 2 and three boundary components are visualised in the linked sections but the examples there have a slightly different symmetry group: the antiprismatic group of order 12.