The derived-groups conjecture for lattice cohomology of plumbed 3-manifolds

Let $Y$ be a plumbed 3-manifold with at most two bad vertices, and let $s$ be a self-conjugate spin$^c$ structure on $Y$. The derived groups $A'$ and $A”$ in the Gysin sequence for $(-Y,s)$ are defined from Pin$(2)$-equivariant monopole Floer homology, while $B'$ and $B”$ are the derived groups in the Gysin sequence of spaces for the lattice cohomology of $Y$.

Derived-groups conjecture. The groups $A'$ and $A”$ are isomorphic, up to a grading shift, to the groups $B'$ and $B”$, respectively.

This conjecture proposes that the derived groups arising from Pin$(2)$-equivariant monopole Floer homology agree with the corresponding groups constructed from lattice cohomology for plumbed 3-manifolds with at most two bad vertices. It is presented as a computation in the general case, and no resolution is supplied here.