Sutured ECH–sutured Floer homology conjecture

Let $(M,\Gamma,\xi)$ be a sutured contact $3$-manifold, let $A\in H_1(M)$, and let $\mathfrak{s}_\xi$ denote the relative $\operatorname{Spin}^c$ structure determined by $\xi$.

Sutured ECH–SFH conjecture. There should be an isomorphism

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

as relatively graded $\mathbb{F}$-modules.

This strengthens the closed ECH–Heegaard Floer conjecture and is supported by calculations in some examples, including sutured contact solid tori with universally tight contact structures. The general statement remains open.