Realization conjecture for singularity categories of finite-dimensional graded algebras

Let $\Lambda$ be a finite-dimensional graded algebra. Write $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ for the dg enhancement of the bounded derived category of an exact category $\mathcal{B}$, let $H$ act on it, and let $\operatorname{pretr}(\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})/H)$ denote the pretriangulated hull of the dg orbit category. The category $\mathcal{B}$ is required to have finite homological dimension, and $\mathsf{D}_{\mathrm{sg}}(\Lambda\textup{-mod})$ denotes the singularity category.

Realization conjecture. There exist an exact category $\mathcal{B}$ of finite homological dimension and a group $H$ acting on $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ such that there is a triangulated equivalence

$$\mathrm{H}^{0}\!\bigl(\operatorname{pretr}(\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})/H)\bigr)\cong\mathsf{D}_{\mathrm{sg}}\!\bigl(\Lambda\textup{-mod}\bigr).$$

This conjecture proposes an analogous dg-orbit realization for the singularity category of every finite-dimensional graded algebra. It is motivated by the paper's discussion of realization results and general duality phenomena; the resolution of the conjecture is not given here.