Analytic order conjecture for isolated critical-point germs

Let $n\ge 3$ and let $f:({\mathbb C}^n,0)\to({\mathbb C},0)$ be a germ of a holomorphic function with an isolated critical point, so that $\mu(f)<\infty$. Let $d(f)$ denote the minimal degree of a polynomial in ${\mathbb C}[x_1,\ldots,x_n]$ right equivalent to $f$ at the origin. Analytic order conjecture. There exists a sequence of positive numbers $a_n$, $n\ge 3$, such that

$$d(f)\le a_n\sqrt[n]{\mu(f)}$$

for every such germ $f$. The classical bounds are $\sqrt[n]{\mu(f)}+1\le d(f)\le\mu(f)+1$, and the paper proves the asserted estimate for germs of type $A_k$; the general higher-dimensional case is left open.