Sutured monopole–Heegaard–instanton homology conjecture

Let $(M,b3)$ be a balanced sutured manifold. Denote its monopole, Heegaard, and instanton sutured homology groups by $aSHM(M,b3)$, $aSFH(M,b3)$, and $aSHI(M,b3)$, respectively. Sutured homology comparison conjecture. The monopole and Heegaard groups $aSHM(M,b3)$ and $aSFH(M,b3)$ are isomorphic. After tensoring with $b3$, they are both isomorphic to the instanton version $aSHI(M,b3)$. This is the principal comparison question motivating the paper; the source gives no evidence of a proof or disproof.