The stratification closure-maps conjecture for quantized coordinate rings

Let $A$ and $H$ be as in Theorem \textup{(c)}. For $J\subset J'$ in $H\text{-}\operatorname{Spec} A$, let $\phi^p_{JJ'}$ and $\phi^s_{JJ'}$ be the maps between closed subsets of the primitive and prime strata, respectively. Stratification closure-maps conjecture. There exist an affine variety $V_{JJ'}$, a corresponding scheme $\widetilde V_{JJ'}$, and morphisms

$$\operatorname{Prim}_{J'} A \to V_{JJ'},\qquad \operatorname{Prim}_J A \to V_{JJ'},\qquad \operatorname{Spec}_{J'} A \to \widetilde V_{JJ'},\qquad \operatorname{Spec}_J A \to \widetilde V_{JJ'}$$

with the displayed maps denoted by $f_{JJ'}$, $g_{JJ'}$, $\widetilde f_{JJ'}$, and $\widetilde g_{JJ'}$, such that

$$\phi^p_{JJ'}(Y)=f_{JJ'}^{-1}\bigl(\overline{g_{JJ'}(Y)}\bigr)$$

for $Y\in {\rm{CL}}(\operatorname{Prim}_J A)$, and

$$\phi^s_{JJ'}(Z)=\widetilde f_{JJ'}^{-1}\bigl(\overline{\widetilde g_{JJ'}(Z)}\bigr)$$

for $Z\in {\rm{CL}}(\operatorname{Spec}_J A)$. The conjecture seeks a classical-geometric description of the relations between strata in the prime and primitive spectra; the immediately preceding discussion notes that simpler descriptions by inverse images or closures of images under morphisms do not always work.