Kontsevich–Soibelman motivic integrality conjecture

Let $\mathcal{A}$ be a category satisfying conditions (7) and (8), let $W$ be its potential, and let $\phi^{mot}$ denote the motivic vanishing-cycle realization. For each dimension vector $d\in\mathbb{N}^{\oplus I}$, the motivic Donaldson–Thomas function $\mathcal{DT}(\mathcal{A},W^*(\phi^{mot}))_d$ is defined in the localized Grothendieck group associated with the critical locus $\mathcal{M}^{W}_d$. Motivic integrality conjecture. For all dimension vectors $d\in\mathbb{N}^{\oplus I}$, the element $\mathcal{DT}(\mathcal{A},W^*(\phi^{mot}))_d$ lies in the image of

$$\operatorname{K}^{\hat{\mu}}(\operatorname{Sch}^{ft}_{\mathcal{M}^{W}_d})\langle \mathbb{L}^{-1/2}\rangle^- \longrightarrow \operatorname{K}^{\hat{\mu}}(\operatorname{Sch}^{ft}_{\mathcal{M}^{W}_d})\langle \mathbb{L}^{-1/2},(\mathbb{L}-1)^{-1}\rangle^-.$$

This conjecture, attributed to M. Kontsevich and Y. Soibelman, asserts motivic integrality: the Donaldson–Thomas functions can be realized without inverting $\mathbb{L}-1$ in the non-stacky motivic theory.