The conjecture on moduli spaces of semiorthogonal decompositions for families of dg categories

Let $\mathcal{D}$ be a family of smooth and proper dg categories over an excellent scheme $U$. For a quasicompact and semiseparated scheme $V$ with a morphism $\phi\colon V\to U$, write $\mathcal{D}_V$ for the base change of the family and let a $V$-linear semiorthogonal decomposition be a decomposition of $\operatorname{Perf}(\mathcal{D}_V)$ into admissible $V$-linear subcategories.

Moduli-space conjecture. There exists an étale algebraic space $\operatorname{\mathsf{SOD}}_{\mathcal{D}/U}$ over $U$ with a functorial bijection

$$\operatorname{\mathsf{SOD}}_{\mathcal{D}/U}(\phi)\simeq\left\{\begin{array}{c}V\text{-linear semiorthogonal}\text{decompositions }\operatorname{Perf}(\mathcal{D}_V)=\langle\mathcal{A},\mathcal{B}\rangle\end{array}\right\}$$

for every such $\phi\colon V\to U$.

This predicts that semiorthogonal decompositions in smooth and proper dg-category families form an étale algebraic moduli space, extending the previously constructed moduli space for families arising from schemes. The conjecture is presented as a generalization suggested by the algebraicity results established earlier in the paper; its resolution is not specified in the supplied text.