Integer multiplicity current conjecture for Seiberg–Witten blow-up sets

A blow-up set $Z$ for the Seiberg–Witten equations with two spinors on a three-dimensional manifold is a closed subset arising as the blow-up set of a sequence of solutions. An integer multiplicity rectifiable current is a rectifiable current equipped with an integer-valued multiplicity function, and its homology class is the associated class in the homology of the ambient manifold. Integer multiplicity current conjecture. Any blow-up set for the Seiberg–Witten equations with two spinors in dimension three admits a structure of an integer multiplicity rectifiable current, whose homology class is the Poincaré dual to the first Chern class of the determinant line bundle. This is motivated by the known local graph-like structure of blow-up sets and by results showing that, on an open everywhere dense subset, they are locally contained in Lipschitz graphs; the proposed current structure would additionally provide an integer multiplicity and an orientation with the specified homology class.