Topological invariance conjecture for sutured embedded contact homology

Let $(M,Γ,ξ)$ be a sutured contact $3$-manifold, let $α_i$ be contact forms with $ξ_i=\ker \alpha_i$, let $J_i$ be almost complex structures, and let $A_1,A_2\in H_1(M)$ be related by

$$A_2-A_1=\operatorname{PD}(\mathfrak{s}_{\xi_1}-\mathfrak{s}_{\xi_2}).$$

Here $\mathfrak{s}_{\xi_i}$ denotes the $\operatorname{Spin}^c$ structure determined by $ξ_i$.

Sutured ECH invariance conjecture. The sutured embedded contact homologies satisfy

$$ECH(M,\Gamma,\alpha_1,J_1,A_1)\simeq ECH(M,\Gamma,\alpha_2,J_2,A_2)$$

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

This conjecture asserts independence from the contact form, contact structure, and almost complex structure. It is motivated by topological invariance of closed ECH, which follows from its identification with Seiberg–Witten Floer cohomology; the sutured case remains conjectural.