Existence of a perfect self-healing cellular automaton (χ(n)=1 for all n)

Determine whether there exists any binary cellular automaton rule, possibly with a neighborhood larger than the 3×3 Moore neighborhood, whose healing efficiency χ(n) equals 1 for every n≥1; alternatively, prove that no such rule exists. Here, for a given rule, χ(n) denotes the fraction of the 2^{n^2} possible n×n square damages to the perfect checkerboard configuration that are fixed under repeated iterations of the rule.

Background

After showing that the checkerboard voting rule’s efficiency χ(n) tends to zero as n grows, the authors pose a broader question about the existence of any rule—regardless of neighborhood size—that can heal all possible finite square damages (i.e., achieve χ(n)=1 for all n). They indicate their suspicion that no such rule exists and note that work on this problem is ongoing, highlighting it as an unresolved research direction.