The combinatorial model conjecture for sutured Legendrian contact homology

Let $\Lambda \subset (\Sigma \times \mathbb{R}, \xi_\Gamma)$ be a Legendrian knot. Let $Y(\Sigma, \Gamma) \subset \Sigma \times \mathbb{R}$ be the convex sutured contact manifold associated to $(\Sigma, \Gamma)$ such that $\Lambda \subset Y(\Sigma, \Gamma)$. Denote by $\mathcal{A}_{\text{red}}(Y, \Lambda)$ the sutured Legendrian contact homology differential graded algebra and by $\widehat{\mathcal{A}}$ the combinatorially defined differential graded algebra constructed in this paper. Combinatorial model conjecture. The dg-algebra $\mathcal{A}_{\text{red}}(Y, \Lambda)$ is quasi-isomorphic to $\widehat{\mathcal{A}}$. The conjecture would identify the analytic sutured Legendrian contact homology algebra with the proposed combinatorial model; the preceding discussion notes that the differential on the generators associated to $\Gamma$ may differ from the actual symplectic-field-theoretic differential because certain holomorphic curves are not currently conceptually excluded.