Bobadilla's conjecture on the smoothness of an irreducible singular locus

Let $f$ be a function germ, let $\Sigma f$ be its singular locus, and let $F_{f,\mathbf 0}$ and $F_{f,\mathbf p}$ denote the Milnor fibers at $\mathbf 0$ and at points $\mathbf p\in\Sigma f$ near $\mathbf 0$, respectively. Assume that $\Sigma f$ is an irreducible curve at $\mathbf 0$ and that

$$H^*(F_{f,\mathbf 0})\cong H^*(F_{f,\mathbf p})$$

for $\mathbf p\in\Sigma f$ near $\mathbf 0$.

Bobadilla's conjecture. Under these assumptions, $\Sigma f$ is smooth at $\mathbf 0$.

This conjecture asks whether constancy of the cohomology of the Milnor fiber along an irreducible one-dimensional singular locus forces that locus to be smooth. The supplied text gives no resolution status.