Effective asymmetric coloring conjecture for inverse limits of finite permutation groups

Let $\mathscr{Gr}$ denote the class of finite permutation groups. Let $(G_i,\varphi_i)_{i\leq k}$ be an epimorphic inverse sequence of length $k$ of finite permutation groups with pairwise disjoint domains. A coloring is zero-neutral and zero-asymmetric in the sense used for the inverse-limit action in the paper.

Effective asymmetric coloring conjecture. There exists a function $g:\mathscr{Gr}\to\mathbb N$ such that, whenever $k\geq g(G_0)$, the inverse limit of this system admits a zero-neutral zero-asymmetric 2-coloring.

The preceding finite theorem proves a bound depending on an intermediate group $G_c$; this conjecture asks for a bound depending only on the initial group $G_0$. The given text does not state whether the conjecture has been resolved.