Compatibility conjecture for structured decompositions of persistent narratives

Let $\textup{\textsf{T}}$ be a finite discrete time category, let $(\mathcal{C},\mathcal{G},\Omega)$ be a spined sd-category with the additional structure required for persistent narratives, and write $\mathfrak{D}(\mathcal{C})$ for the category of structured decompositions in $\mathcal{C}$. Let $\mathfrak{D}(\textup{\textsf{Pe}}(\textup{\textsf{T}},\mathcal{C}))$ denote the category of structured decompositions of persistent narratives and $\textup{\textsf{Pe}}(\textup{\textsf{T}},\mathfrak{D}(\mathcal{C}))$ the category of persistent narratives of structured decompositions. Compatibility conjecture. Under appropriate assumptions, there exists a fully faithful functor, injective on objects,

$$\mathfrak{D}(\textup{\textsf{Pe}}(\textup{\textsf{T}},\mathcal{C})) \hookrightarrow \textup{\textsf{Pe}}(\textup{\textsf{T}},\mathfrak{D}(\mathcal{C})).$$

Such an embedding would formalize the compatibility between forming structured decompositions and forming persistent narratives, allowing concepts and results for one construction to be transferred to the other. The source leaves the appropriate assumptions and the existence of this functor as an open question.