Construct analogous examples with an embedded regular part

Construct stationary integral varifolds with a flat tangent plane of multiplicity two at an isolated singular point and infinite topology accumulating at that point whose regular minimal surface is embedded outside the singular point.

Background

The constructed varifold is smooth but immersed away from its singular point; in the final construction, the neck regions are explicitly non-embedded. The paper discusses a more sophisticated catenoidal construction that might preserve embeddedness, but does not establish it.

The second explicitly stated open question is whether similar examples can be made embedded outside the singular point. Resolving it would demonstrate that accumulation of infinite topology at a flat singularity is possible even without self-intersections in the regular part.