Kontsevich–Soibelman motivic integration map conjecture

Let $\mathcal{A}$ be the category considered above, with $L$ its grading lattice, $\Lambda$ the coefficient ring, and $\mathrm{MH}(\mathcal{A})$ and $\mathrm{QT}(\mathcal{A})$ the associated motivic Hall algebra and quantum torus. The map is required to be defined by taking motivic invariants.

Kontsevich–Soibelman integration-map conjecture. There exists an $L$-graded $\Lambda$-algebra homomorphism

$$I_{\mathrm{KS}}\colon \mathrm{MH}(\mathcal{A})\to \mathrm{QT}(\mathcal{A})$$

This is the motivic precursor of the integration maps later constructed in the semi-classical setting. The supplied text does not establish the conjecture unconditionally: the cited construction is conditional on an expected formula for motivic Milnor fibers.