Fernández de Bobadilla's conjecture on non-splitting of Milnor-fiber cohomology

Let $\mathcal{U}$ be an open neighborhood of the origin in $\mathbb{C}^{n+1}$, let $f:(\mathcal{U},\mathbf{0})\to(\mathbb{C},0)$ be a complex analytic function with $\dim_{\mathbf{0}}\Sigma f=1$, and choose coordinates $\mathbf{z}=(z_0,\ldots,z_n)$ such that $\dim_{\mathbf{0}}\Sigma(f|_{V(z_0)})=0$. Let $\Sigma f$ be the critical locus of $f$, let $F_{f,\mathbf{0}}$ be the Milnor fiber of $f$ at the origin, and for each irreducible component $C$ of $\Sigma f$ at $\mathbf{0}$ let $\mu^\circ_C$ denote the Milnor number of a generic transverse slice along $C$. Fernández de Bobadilla's conjecture. If

$$\widetilde H^*(F_{f,\mathbf{0}};\mathbb{Z})$$

is non-zero only in degree $n-1$, and

$$\widetilde H^{n-1}(F_{f,\mathbf{0}};\mathbb{Z})\cong\bigoplus_C\mathbb{Z}^{\mu^\circ_C},$$

then $\Sigma f$ has a single irreducible component, which is smooth. This extends the GLL non-splitting phenomenon from a hyperplane slice to the Milnor fiber of $f$ itself; the supplied text gives no resolution status.