The manifold-with-corners conjecture for augmented stability conditions

Let $\sigma = \langle \mathcal{C}_\bullet \mid \sigma_\bullet \rangle_\Sigma$ be an augmented stability condition on a smooth and proper stable $dg$-category $\mathcal{C}$ over a field. Suppose that for every pair of terminal vertices $u,v \in \Gamma(\Sigma)$ with $\mathcal{C}_{\leq u} \subseteq \mathcal{C}_{\leq v}$, the inclusion functor admits either a right or a left adjoint. For $\sigma$-stable objects $E_1,\ldots,E_n \in \mathcal{C}$ whose classes $v(E_i)$ form a basis for $\Lambda \otimes \mathbb{Q}$, let $\ell_{(-)}(E_1,\ldots,E_n): \mathcal{A}\operatorname{Stab} \to \mathcal{A}^{\mathbb{R}}_n$ be the log-central-charge map.

Manifold-with-corners conjecture. The map $\ell_{(-)}(E_1,\ldots,E_n)$ induces a homeomorphism from an open neighborhood of $\sigma$ onto an open subset of $\mathcal{A}^{\mathbb{R}}_n$.

This is the simple, local form of the conjecture that the space of augmented stability conditions has a manifold-with-corners structure. The paper proves the assertion at generic admissible points, while the corresponding statement at non-generic admissible boundary points remains open.