Monopole classes of connected sums of 3-manifolds

Let $N_i$ for $i=1,\ldots,n$ be closed oriented $3$-manifolds, and let $N_1\#\cdots\#N_n$ denote their connected sum. A monopole class is a second cohomology class arising as the first Chern class of a $\operatorname{Spin}^c$ structure whose Seiberg–Witten equations admit a solution for every Riemannian metric. The rational part of a cohomology class means its component in the rational cohomology considered by the source.

Connected-sum monopole-class conjecture. The rational part of a monopole class of $N_1\#\cdots\#N_n$ is expressed as

$$\sum_{i=1}^n\alpha_i,$$

where $\alpha_i$ is a monopole class of $N_i$.

This conjecture proposes that monopole classes on a connected sum decompose into contributions from the individual summands, at least after taking the rational part. The source gives no resolution status.