The $\operatorname{Stab}^d$ conjecture for truncated stability-condition spaces

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$ and let $d\geq0$. Let $\mathfrak U_n^{>d}$ be the subspace of central charges defined by the interlacing conditions and the separation bound $\operatorname{sep}(\ell(\underline s,\underline t))>d$. $\operatorname{Stab}^d$ conjecture. There exists a family of stability conditions $\operatorname{Stab}_H^{*>d}(X)$ with respect to the $H$-polarized lattice $\Lambda_H$ such that

$$\operatorname{Forg}:\operatorname{Stab}_H^{*>d}(X)\to\operatorname{Hom}(\Lambda_H,\mathbb C),\qquad \sigma=(\mathcal A,Z)\mapsto Z$$

is a homeomorphism onto $\mathfrak U_n^{>d}$, and tensoring by $\mathcal O_X(H)$ preserves the family. This is one of the paper's main conjectures, refining the $d=0$ standard-slice assertion by imposing a separation threshold; no resolution is given in the supplied text.