The sutured embedded contact homology–sutured Floer homology isomorphism conjecture

Let $(M,\Gamma,\alpha)$ be a sutured contact manifold. Here $ECH(M,\Gamma,\alpha)$ denotes sutured embedded contact homology and $SFH(-M,-\Gamma)$ denotes sutured Floer homology of the orientation-reversed sutured manifold.

Sutured ECH–SFH isomorphism conjecture. There is an isomorphism

$$ECH(M,\Gamma,\alpha)\cong SFH(-M,-\Gamma),$$

which moreover respects the decomposition into first homology classes on the left-hand side and relative Spin$^c$-structures on the right-hand side.

Sutured embedded contact homology was defined in analogy with sutured Floer homology, and this conjecture proposes that the two theories agree, including their respective gradings or decompositions. Its resolution is not specified in the source.