Relative monodromy conjecture for principal monodromy

Let $F$ be a number field, let $K$ be a completion of $F$, and let $Z_{\Phi}(s,\chi,f_l,V_{l-1},K)$ be the local zeta function associated with the complex non-degenerate polynomial mapping described above. Let $f_l$ define the Milnor fibration on the complete intersection $V^{(l-1)}$, and call its monodromy relative to $V^{(l-1)}$ at the origin the $l$-th principal monodromy. Relative monodromy conjecture. For almost all the completions $K$ of $F$, if $s$ is a pole of $Z_{\Phi}(s,\chi,f_l,V_{l-1},K)$, then

$$\exp\left(2\pi\sqrt{-1}\operatorname{Re}(s)\right)$$

is an eigenvalue of the $l$-th principal monodromy of $f_l$ relative to $V^{(l-1)}$ at the origin. This conjecture predicts that the real parts of poles of the local zeta function determine eigenvalues of the corresponding principal monodromy, extending the classical relationship between poles of local zeta functions and monodromy in the non-degenerate setting.