Mond conjecture for image Milnor and -codimension numbers

Let $f:(\mathbb{C}^n,S)\rightarrow (\mathbb{C}^{n+1},0)$ be an $\mathscr{A}$-finite mapping, where $(n,n+1)$ are Mather's nice dimensions. The image Milnor number $\mu_I(f)$ and the $\mathscr{A}_e$-codimension $\operatorname{codim}_{\mathscr{A}_e}(f)$ are the corresponding invariants for the image hypersurface.

Mond conjecture.

$$\mu_I(f)\geq \operatorname{codim}_{\mathscr{A}_e}(f),$$

with equality if $f$ is weighted homogeneous.

The conjecture extends the inequality between the Milnor and Tjurina numbers to images of mappings with isolated instability. The only known cases are $n=1,2$, and it remains open whether the inequality and equality statement hold for $n\geq 3$.