Polylogarithmic bound for effective asymmetric coloring of inverse limits

Let $\mathscr{Gr}$ be the class of finite permutation groups, and let $\operatorname{asy}(G)$ denote the asymmetric coloring number of a finite permutation group $G$. Let $g$ be a function as in the effective asymmetric coloring conjecture: for every epimorphic inverse sequence of finite permutation groups with disjoint domains, length at least $g(G_0)$ guarantees a zero-neutral zero-asymmetric 2-coloring of the inverse limit.

Polylogarithmic bound conjecture. There exists a polynomial $p$ such that the effective asymmetric coloring conjecture holds with

$$g(G)=p\bigl(\log(\operatorname{asy}(G))\bigr).$$

The preceding argument gives the lower bound $g(G)\geq\log_2\operatorname{asy}(G)$ whenever the effective conjecture holds. This conjecture asks whether a polynomial in the logarithm gives a corresponding upper bound; its status is not given in the supplied text.