Generalized Fernández de Bobadilla conjecture on critical loci

Let $f$ be a holomorphic function germ with critical locus $\boldsymbol{\Sigma} f$, let $F_{f,\mathbf 0}$ denote its Milnor fiber at the origin, and let $\stackrel{\circ}{\mu}_C$ be the transverse Milnor number associated with each irreducible component $C$ of $\boldsymbol{\Sigma} f$. Generalized Fernández de Bobadilla conjecture. Suppose that

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

is nonzero only in degree $n-1$ and

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

where the sum is over all irreducible components $C$ of $\boldsymbol{\Sigma} f$. Then $\boldsymbol{\Sigma} f$ has a single irreducible component, which is smooth. The source introduces this as the natural generalization of Fernández de Bobadilla's conjecture; no resolution is supplied.