Non-formality in $\mathrm{P}\mathrm{i}\mathrm{n}(2)$-monopole Floer homology

In previous work, we introduced a natural ${\mathcal{A}}_{\mathrm{\infty }}$-structure on the $\mathrm{P}\mathrm{i}\mathrm{n}(2)$-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we provide explicit descriptions of several Massey products induced on homology, and discuss how they can be used to compute the $\mathrm{P}\mathrm{i}\mathrm{n}(2)$-monopole Floer homology of connected sums in many concrete examples.