The adjacency injection conjecture for discriminant-complement groups

Let $f$ and $g$ be singular functions such that $f$ is adjacent to $g$, so that a versal unfolding of $g$ is also a versal unfolding for $f$. Let $4mu$ and $tilde\mu$ be the corresponding Milnor numbers, and write ${\cal D}_f$ and ${\cal D}_{\tilde f}$ for the discriminant complements in the respective parameter spaces. Adjacency injection conjecture. The natural map

$$\pi_1({\mathbf C}^{\mu}\setminus{\cal D}_f)\longrightarrow \pi_1({\mathbf C}^{\tilde\mu}\setminus{\cal D}_{\tilde f})$$

is injective, fitting into the commutative diagram with the free groups $F_\mu$ and $F_{\tilde\mu}$ displayed in the source. This conjecture concerns how fundamental groups behave under adjacency of singular functions; the source presents it as a speculation and gives no resolution.