Sufficiency of the gcd-at-most-two condition for generic homogeneous germs

Determine whether generic homogeneous corank-one germs with degrees [?]mu and k satisfying gcd([?]mu,k)[?]\\le2 are [?]A-finite in every source dimension.

Background

The paper proves [?]A-finiteness for a dense open set of homogeneous pairs of coprime degrees. The coprimality assumption is used to exclude multiple points on the punctured coordinate axis. The authors explain that when the degree gcd is at least three, generic branches fail to meet in general position and the germ cannot be [?]A-finite.

For gcd equal to two, the authors report computational examples that are [?]A-finite in selected dimensions and degree pairs, but they do not establish a general theorem. The unresolved question is whether gcd at most two suffices in general.

References

We do not know whether $\gcd(\mu,k)\le2$ suffices in general.

— Mond's conjecture for corank-one maps from $\mathbb C^3$ to $\mathbb C^4$  (2610.03331 - Rimanyi, 2 Oct 2026) in Remark 3? “viability” in Section 8, “Beyond quasihomogeneity: families of unbounded corank-one multiplicity”