The manifold-with-corners conjecture at admissible points

Let $\mathcal{C}$ be a stable $dg$-category, and let $\sigma = \langle \mathcal{C}_\bullet \mid \ell_\bullet \rangle_\Sigma \in \mathcal{A}\operatorname{Stab}(\mathcal{C})$ be an admissible point. Let $P_1,\ldots,P_n \in \mathcal{C}$ be $\sigma$-stable objects, where $n=\operatorname{rank}(\Lambda)$, such that for each terminal vertex $v$, the vectors $v(P_i)$ among those $P_i$ with $\operatorname{dom}(P_i)=v$ span $\Lambda_v \otimes \mathbb{Q}$. Consider the map $\ell_{(-)}(P_1,\ldots,P_n):U \to \mathcal{A}^{\mathbb{R}}_n$.

Manifold-with-corners conjecture. There is an open neighborhood $U$ of $\sigma$ such that this map is a homeomorphism from $U$ onto an open subset of $\mathcal{A}^{\mathbb{R}}_n$.

The paper proves this assertion at every admissible point that is also generic. The full assertion for admissible, non-generic boundary points is not resolved in the supplied text.