The multivariate topological monodromy conjecture

Let $F=(f_1,\dots,f_k)$ be a $k$-tuple of nonconstant polynomials in $\mathbb{C}[x_1,\ldots,x_n]$. Let $Z_{\operatorname{top}}(F;\mathbf{s})$ be its multivariate topological zeta function, let $\mathcal{P}(Z_{\operatorname{top}}(F;\mathbf{s}))$ be the support of its polar divisor, and let $\mathcal{S}_F\subset(\mathbb{C}^*)^k$ be the monodromy support, consisting of the parameters $\alpha$ for which some local complement has nonzero cohomology with coefficients in the associated rank-one local system. Define

$$\operatorname{Exp}:\mathbb{C}^k\to(\mathbb{C}^*)^k,\qquad \operatorname{Exp}(\beta_1,\dots,\beta_k)=(e^{2\pi i\beta_1},\dots,e^{2\pi i\beta_k}).$$

The multivariate monodromy conjecture. If $F$ is such a tuple, then

$$\operatorname{Exp}\left(\mathcal{P}\left(Z_{\operatorname{top}}(F;\mathbf{s})\right)\right)\subset\mathcal{S}_F.$$

This extends the one-polynomial monodromy conjecture from individual poles and eigenvalues to polar hyperplanes and the monodromy support of a tuple; the source presents it as a conjectural multivariate generalization.