Denef–Sargos polar multiplicity conjecture for stable faces

Let $f$ be a singular complex analytic germ at $0\in\mathbb{C}^{n}$ with $f(0)=0$, and let $\varphi$ be a Schwarz function on $\mathbb{C}^{n}$ with sufficiently small support and $\varphi(0)\neq 0$. Let $Z(s)$ be the complex local zeta function associated to $f$ and $\varphi$, let $s_{0}\notin\mathbb{Z}$, and let $\tau_{0}$ be the relevant face of the Newton polyhedron. The polar multiplicity of $Z(s)$ at $s_{0}$ is the order of its pole there, and $\tau_{0}$ is a stable face when it satisfies the stability condition of Denef and Sargos. Denef–Sargos polar multiplicity conjecture. The polar multiplicity of $Z(s)$ at $s_{0}$ equals $\rho$ if and only if $\tau_{0}$ is a stable face. The source presents this as the complex stability conjecture; no resolution is supplied in the excerpt.