Relative quasi-convergence and compactification of the stability manifold

Let $R$ be a ring and let $\mathcal{D}$ be an $R$-linear, smooth and proper category. A continuous path $\sigma_\bullet:[0,+\infty)\rightarrow\operatorname{Stab}(\mathcal{D}/R)$ of locally constant stability conditions is relative quasi-convergent if, for every $s\in\operatorname{Spec} R$, the corresponding path $\sigma_{s,\bullet}$ is quasi-convergent. Relative quasi-convergence conjecture. A relative quasi-convergent path induces an equivalence relation $\sim^i$ on $\mathcal{D}$ and a corresponding $R$-linear semiorthogonal decomposition

$$\mathcal{D}=\langle\mathcal{D}_1,\dots,\mathcal{D}_m\rangle.$$

Moreover, the complex manifold $\operatorname{Stab}(\mathcal{D}/R)^\circ$ admits a compactification $\mathcal{A}\operatorname{Stab}(\mathcal{D}/R)$ by locally constant augmented stability conditions for which the quasi-convergent paths in $\operatorname{Stab}(\mathcal{D}/R)$ are convergent. The conjecture proposes a relative version of the decomposition extracted from quasi-convergent paths in the absolute setting, together with a compactification by augmented stability conditions; the source notes that it is unclear whether additional constraints on relative quasi-convergence are required.