The reduced-stability conjecture

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$, and let $\Lambda_H$ be the $H$-polarized lattice. Let $\mathfrak B_n$ and $\pm\mathfrak B_n$ be the spaces of reduced central charges defined from the functionals $\mathsf B_{\underline t}$, and write $\operatorname{Sb}_H^*(X)$ for reduced stability conditions. Reduced-stability conjecture. There exists a family of reduced stability conditions $\operatorname{Sb}_H^*(X)$ such that

$$\operatorname{Forg}:\operatorname{Sb}_H^*(X)\to\operatorname{Hom}(\Lambda_H,\mathbb R),\qquad \widetilde\sigma=(\mathcal A,B)\mapsto B$$

is a homeomorphism onto $\mathfrak B_n$, the extended map

$$\operatorname{Forg}:\coprod_{n\in\mathbb Z}\operatorname{Sb}_H^*(X)[n]\to\pm\mathfrak B_n$$

is a universal cover, the family is preserved by tensoring with $\mathcal O_X(H)$, and whenever $\underline s<\underline t<\underline s[1]$ one has $\widetilde\sigma_{\underline s}\lesssim\widetilde\sigma_{\underline t}\lesssim\widetilde\sigma_{\underline s}[1]$. These properties generalize the corresponding constructions known for curves, surfaces, and certain threefolds; the conjecture remains open in the general setting considered here.