Denef–Sargos instability conjecture for oscillating integrals

Let $f$ be a real analytic germ at $0\in\mathbb{R}^{n}$ with Newton polyhedron, and let $\tau_{0}$ be the face determining the relevant pole and coefficient $\mu(\varphi)$ of the associated oscillating integral. The face $\tau_{0}$ is unstable when it has the instability property defined in the paper. Denef–Sargos instability conjecture. If $\tau_{0}$ is unstable, then $\mu(\varphi)=0$ for any $\mathcal{C}^{\infty}$-function $\varphi$ with support in a sufficiently small neighbourhood of $0$ in $\mathbb{R}^{n}$. This is the conjecture of Denef and Sargos that the paper proves; the converse implication is stated separately as another open conjecture.