Mond’s conjecture in source dimensions at least four

Prove Mond’s conjecture for all [?]A-finite corank-one map germs from [?]C^m to [?]C^{m+1} when m[?], beyond the cases established in the paper.

Background

Mond’s conjecture asserts that the [?]A_e-codimension of a finite-codimension map germ is at most its image Milnor number, with equality for quasihomogeneous germs. The paper proves the conjecture for corank-one germs from [?]C3 to [?]C4 and reduces the higher-dimensional problem to inequalities for explicit families of homogeneous germs.

The introduction explicitly records that the conjecture remains unresolved in dimensions m[?]; the later discussion indicates that the case m=4 still requires a separate lifting argument and an upper bound for generic homogeneous germs.

References

It was formulated more than thirty years ago and remains open for $m\ge3$.

— Mond's conjecture for corank-one maps from $\mathbb C^3$ to $\mathbb C^4$  (2610.03331 - Rimanyi, 2 Oct 2026) in Section 1, Introduction, subsection “Motivation”

In computer experiments with unweighted germs, this recursive structure leads to a conjectural closed formula for $Hilb_tN$.

— Mond's conjecture for corank-one maps from $\mathbb C^3$ to $\mathbb C^4$  (2610.03331 - Rimanyi, 2 Oct 2026) in Section 9, “What changes for m>3?”