Fernández de Bobadilla's smooth critical-locus conjecture

Let $f$ be a holomorphic function germ with critical locus $\boldsymbol{\Sigma} f$, let $C$ be its irreducible component, and let $F_{f,\mathbf p}$ denote the Milnor fiber of $f$ at a point $\mathbf p$. Write $n$ for the ambient complex dimension and let $\stackrel{\circ}{\mu}_C$ be the corresponding transverse Milnor number. Fernández de Bobadilla's conjecture. Suppose that the critical locus of $f$ has a single irreducible component $C$ and that the isomorphism type of the cohomology groups $\widetilde H^*(F_{f,\mathbf p};{\mathbb Z})$ is independent of the choice of $\mathbf p\in C$, with

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

nonzero only in degree $n-1$, and

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

Then $C$ is smooth. The conjecture was presented as related to the Lê–Lazzeri–Gabrielov non-splitting theorem, but does not follow from that theorem by any known argument and remains open.