Igusa–Denef–Loeser motivic monodromy conjecture

Let $f$ define a hypersurface, let $x$ be a point in its domain, and let $Z_{f,x}(T)$ be the local motivic zeta function. Write ${\mathbb L}$ for the Lefschetz motive and let $\mathcal{M}^{\hat{\mu}}_\mathbb{C}$ denote the relevant completed Grothendieck ring with profinite roots-of-unity action. A monodromy eigenvalue of $f$ at a point of $f^{-1}(0)$ means an eigenvalue of the local monodromy on the corresponding local Milnor fiber. Igusa–Denef–Loeser motivic monodromy conjecture. There exists a finite subset $S$ of ${\mathbb Z}_{>0} \times {\mathbb Z}_{>0}$ such that

$$Z_{f,x}(T) \in \mathcal{M}^{\hat{\mu}}_\mathbb{C}\left[T, \frac{1}{1-\mathbb{L}^{-a}T^b}\right]_{(a,b)\in S}$$

and such that for each $(a,b)\in S$, the value $\exp(-2\pi i b/a)$ is a monodromy eigenvalue of $f$ at some point of $f^{-1}(0)$. The conjecture connects the possible motivic-zeta denominators with local monodromy eigenvalues; naive, topological and $p$-adic analogues have been proved in various cases, but the motivic statement as presented here is resolved according to the supplied status.