Sutured Heegaard Floer–embedded contact homology isomorphism conjecture

Let $(M,\Gamma,\xi)$ be a sutured contact $3$-manifold. For $A\in H_1(M;\mathbb{Z})$, let $\mathfrak{s}_\xi$ be the canonical Spin$^c$-structure determined by $\xi$, let $ECH(M,\Gamma,\xi,A)$ denote the sutured embedded contact homology in homology class $A$, and let $SFH(-M,-\Gamma,\mathfrak{s}_\xi+\operatorname{PD}(A))$ denote the sutured Heegaard Floer homology in the indicated Spin$^c$-structure. Sutured Heegaard Floer–embedded contact homology isomorphism conjecture. There is an isomorphism of relatively graded vector spaces over $\mathbb{Z}/2\mathbb{Z}$

$$ECH(M,\Gamma,\xi,A)\simeq SFH(-M,-\Gamma,\mathfrak{s}_\xi+\operatorname{PD}(A)).$$

Moreover, this isomorphism identifies the contact invariant of $\xi$ in $ECH(M,\Gamma,\xi,0)$ with the contact invariant in $SFH(-M,-\Gamma,\mathfrak{s}_\xi)$. The conjecture is a slight strengthening of a prior conjecture relating sutured Heegaard Floer homology and embedded contact homology; the paper presents the equivalence of the sutured theories as a conjectural relation in this setting, alongside established equivalences involving sutured monopole Floer homology.