Loeser's generalized monodromy conjecture for multivariable zeta functions

Let $K$ be a number field, let $X$ be a smooth algebraic variety over $K$, and let $F=(f_1,\ldots,f_r):X\to\mathbb{A}^r_K$. For almost all finite places $\mathfrak P$, let $F_{\mathfrak P}$ be the induced analytic map and let $Z_\phi(F_{\mathfrak P},\chi;s)$ be its multivariable Igusa zeta function, with $\phi$ residual. Fix an identification $\overline{\mathbb{Q}}_l\simeq\mathbb{C}$. For an affine hyperplane $H$ with equation $\sum_jN^js_j+\nu=0$, define $\mathcal H$ as the Zariski closure of the corresponding finite-order characters. Let $\mathcal A^q_{F,\xi}(\overline{\mathbb{Q}}_l)$ denote the relevant support associated with nearby-cycle data at a geometric point $\xi$.

Loeser's generalized monodromy conjecture. For almost all finite places $\mathfrak P$, if $H$ is a polar hyperplane in $\mathcal P(Z_\phi(F_{\mathfrak P},\chi;s))$, with $\phi$ residual on $X_{\mathfrak P}$, then, for every irreducible component $Z$ of $\mathcal H$, there exists an integer $q$ and a geometric point $\xi$ such that $Z$ is contained in the support of $\mathcal A^q_{F,\xi}(\overline{\mathbb{Q}}_l)$.

This generalizes the one-variable monodromy conjecture by relating every component arising from a polar hyperplane to the support of multivariable nearby-cycle data. The supplied source does not state whether the conjecture is open, proved, or refuted.