The standard-slice conjecture for Bridgeland stability conditions

Let $(X,H)$ be an $n$-dimensional irreducible smooth polarized variety over $\mathbb{C}$. Let $\Lambda_H$ be the $H$-polarized lattice, and let $\mathfrak U_n$ be the specified open subset of $\operatorname{Hom}(\Lambda_H,\mathbb{C})$ defined by the interlaced-polynomial construction. Write $\operatorname{Stab}_H(X)$ for stability conditions on $D^b(X)$ with respect to $\Lambda_H$. Standard-slice conjecture. There exists a family $\operatorname{Stab}_H^*(X)$ of stability conditions on $D^b(X)$ such that the forgetful map

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

is a homeomorphism onto $\mathfrak U_n$, and $\operatorname{Stab}_H^*(X)$ is invariant under tensoring by $\mathcal O_X(H)$. This proposes a uniform family of stability conditions for smooth projective varieties; its existence is part of the paper's conjectural framework and is not resolved here.