Motivic monodromy conjecture for local motivic zeta functions

Let $k$ be a field of characteristic zero, let $X$ be the relevant $k$-variety, let $f:X\to\mathbb{A}^1_k$ be the morphism, let $x$ be a closed point with residue field $k_x$, and let $Z_{f,x}(T)\in\mathcal{M}_k[[T]]$ be the local motivic zeta function. Assume that $k$ is a subfield of $\mathbb{C}$. Motivic monodromy conjecture. There exists a finite subset $\mathscr{S}\subset\mathbb{Z}_{<0}\times\mathbb{Z}_{>0}$ such that

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

and such that for every $(a,b)\in\mathscr{S}$, $a/b$ is a root of the Bernstein–Sato polynomial $b_f(s)$. In particular, there exists $y\in X(\mathbb{C})$ with $f(y)=0$ such that ${\mathcal{E}\mathrm{xp}}(2\pi ia/b)$ is a local monodromy eigenvalue of $f$ at $y$. This conjecture is the motivic analogue of the strong $p$-adic monodromy conjecture; the paper notes that the formulation can be extended using $\ell$-adic nearby cycles, while the general conjecture remains open.