Smooth minimal-surface realizations of the constructed varifold

Determine whether the stationary integral varifold constructed in Theorem  or a varifold of the same type can arise as a limit of smooth minimal surfaces, including by resolving its multiplicities into distinct sheets and desingularizing its junction circles.

Background

The paper constructs a stationary integral 2-varifold in $\mathbb{R}^3$ contained in a narrow cone and having volume growth at least $R^{2+\frac45\varepsilon^2}$. The construction is a locally finite weighted union of flat disks, flat annuli, and catenoidal pieces meeting along circles; the weights are positive integer multiplicities, so the resulting object is generally neither smooth nor parameterizable by a single smooth connected immersed surface.

The authors explicitly leave unresolved whether this singular, multiplicity-bearing varifold—or a similar one—can be obtained as a limit of smooth minimal surfaces. They identify two concrete technical requirements for such a construction: replacing multiplicities by distinct sheets and desingularizing every junction circle. They mention bent saddle towers and existing desingularization methods as possible approaches, but do not establish the result.