Motivic Milnor fiber comparison with the topological Milnor fiber

Let $f$ be a complex analytic function in $d$ variables vanishing at the origin $O$. Let $F_{f,O}$ be its topological Milnor fiber, let $\mathscr S_{f,O}$ be the motivic Milnor fiber of a Taylor expansion of $f$ at $O$, and let $\chi_c$ and $\chi_h$ denote respectively the compactly supported Euler characteristic and the Hodge characteristic with monodromy. Motivic Milnor fiber comparison conjecture. The following equalities hold:

$$\chi_c(\mathscr S_{f,O})=\chi_c(F_{f,O}).$$ $$\chi_h(\mathscr S_{f,O})=\chi_h(F_{f,O})\quad\text{in }K_0(\operatorname{HS}^{\operatorname{mon}}).$$

These equalities assert that the motivic Milnor fiber recovers both the ordinary compactly supported Euler characteristic and the Hodge-theoretic invariant of the topological Milnor fiber, including its finite-order monodromy.