Free boundary minimal surfaces of genus one

Two parallel discs which are very close to the equatorial disc can be connected by a sufficiently large number of half-necks along the equator and by one catenoidal neck in the center and then deformed to a free boundary minimal surface in the unit ball which has genus one and a large number of boundary components. It is an open question whether such surfaces also exist with a small number of boundary components as visualised conjecturally below.

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