Denef–Loeser's motivic monodromy conjecture

Let $k$ be a field of characteristic zero, let $X=\mathbb A^d_{\mathbb C}$, let $h\in\mathbb C[x_1,\ldots,x_d]$ satisfy $h(\mathbf 0)=0$, and let $Z_{DL}(h,T)$ be the local Denef–Loeser motivic zeta function. Write $\mathcal M_k$ for the naive motivic ring and $\mathbb L=[\mathbb A_k^1]$. Denef–Loeser's motivic monodromy conjecture. There is a set $S=\{(a,b):a,b\in\mathbb N,\ b>0\}$ such that

$$Z_{DL}(h,T)\in\mathcal M_k[T][(1-\mathbb L^{-a}T^b)^{-1}]_{(a,b)\in S},$$

and, if $q=a/b$ with $(a,b)\in S$, then $\exp(2i\pi q)$ is an eigenvalue of the local complex algebraic monodromy around zero at some $P\in h^{-1}(0)$. This motivic conjecture is linked to the local Igusa and topological zeta-function conjectures; the paper proves the relevant monodromy conjectures for quasi-ordinary power series.