Uniform stability conjecture for vanishing cycles

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p\neq 2$, let $\mathcal{F}\in D(X)$, and let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. For an étale neighbourhood $U$ of $x$, let $f,g:U\to\mathbb{A}^1$ be test functions at $(x,\xi)$, and let $\mathfrak{m}_x$ be the maximal ideal at $x$. The uniform stability conjecture. There exists a positive integer $N$, depending on $(x,\xi)$, such that for every such $U,f,g$ satisfying $f\equiv g\pmod{\mathfrak{m}_x^N}$, there is an isomorphism

$$\phi_f(\mathcal{F})_x\cong\phi_g(\mathcal{F})_x$$

as objects in $D^b_c(\mathbb{Z}/\ell^n[G_{\eta}])$. This conjecture asks for a uniform, representation-theoretic stability statement for vanishing cycles, strengthening the paper's comparison with Saito's result, which fixes a single test function and allows an isolated characteristic point. The conjecture remains open in the stated generality.