Bilinearised boundary model conjecture for exterior complexes

Let $\Lambda$ be a sutured Legendrian with boundary components $\partial_0\Lambda$ and $\partial_1\Lambda$ in $(\Gamma,\lambda_\Gamma)$. Let $LC^{\mathrm{ext}}(\Lambda,\partial_0\Lambda,\partial_1\Lambda;\lambda,J,H)$ be the quotient of the wrapped complex by the cylindrical complex, with coefficients in $H_1(\Lambda)$ induced by the inclusion $\partial\Lambda\hookrightarrow\Lambda$, and let $LC_\varepsilon$ denote the bilinearised boundary complex for the augmentations induced by the relevant Legendrians. Exterior complex conjecture. The exterior complex is quasi-isomorphic to the shifted bilinearised boundary complex

$$LC^{\mathrm{ext}}(\Lambda,\partial_0\Lambda,\partial_1\Lambda;\lambda,J,H)\simeq LC_\varepsilon(\partial_0\Lambda,\partial_1\Lambda;\Gamma,\lambda_\Gamma,J_\Gamma)[1].$$

Furthermore, for a path of sutured Legendrians whose boundary determines a Legendrian loop and whose endpoint components are hypertight, the associated exact triangle is invariant and the diagram of invariance maps commutes; the boundary map is induced by the corresponding Lagrangian cobordism and is the identity when the boundary is fixed. These are formulated as expected results extending the cited theorems, and the source states that they hold trivially in the particular application considered there.