In 1993, David Hoffman and Hermann Karcher considered the possibility of a minimal surface with the appearance shown above. It would be a genus 2 variation of the Costa surface, with the same symmetries.
In the simplest most symmetric case, the Enneper-Weierstrass data contain a parameter that needs to be adjusted to solve a single equation. While the top image is pretty convincing, the one above shows a small gap which gets significantly worse below.
This casts doubt on the first image: Did I really solve the period problem, or is the remaining gap just so small that we don’t see it anymore? The latter is the case, this surface does not exist, i.e. the period problem cannot be solved, at least not in the most symmetric case.
David Hoffman and Hermann Karcher gave the surface the name Horgan surface, to make a case for the necessity of rigorous proof and countering suggestions made by John Horgan in an article in Scientific American that proofs were about to become obsolete, and numerical experiments supplied by computers could be an acceptable substitute.