The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs

With $f_1,f_2$ the Zariski counterexample germs, $\tilde f_j=f_j(z)+t^2$ on $\mathbb{C}^{N+1}$ are topologically right equivalent with isolated critical points, but $H_0((\tilde f_1)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}$ while $H_0((\tilde f_2)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}^2$: one initial-form Milnor fibre is connected and the other has two components. This disproves both the embedded and the right-equivalence formulations. It says nothing about weaker invariants the conjecture was meant to capture, such as cohomology of contact loci.

With $f_1,f_2$ the Zariski counterexample germs, $\tilde f_j=f_j(z)+t^2$ on $\mathbb{C}^{N+1}$ are topologically right equivalent with isolated critical points, but $H_0((\tilde f_1)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}$ while $H_0((\tilde f_2)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}^2$: one initial-form Milnor fibre is connected and the other has two components. This disproves both the embedded and the right-equivalence formulations. It says nothing about weaker invariants the conjecture was meant to capture, such as cohomology of contact loci.