Optimality of the checkerboard voting rule within the 3×3 Moore neighborhood

Determine whether there exists any binary cellular automaton rule with the 3×3 Moore neighborhood that achieves strictly higher healing efficiency χ(n) for n×n square damages to the perfect checkerboard configuration than the checkerboard voting rule F defined in the paper; alternatively, prove that no such 3×3 Moore-neighborhood rule can outperform F. Here, for a given rule, χ(n) denotes the fraction of the 2^{n^2} possible n×n square damages that are fixed (heal) under repeated iterations of the rule.

Background

The paper introduces a binary cellular automaton, the checkerboard voting rule F (topologically conjugate to the majority rule), as a model of self-healing on a two-dimensional checkerboard substrate. It defines the healing efficiency χ(n) as the fraction of fixable n×n square damages to the checkerboard pattern and reports empirical and analytical results for F, including χ(n)=1 for n≤3, high but decreasing χ(n) for moderate n, and χ(n)→0 as n→∞.

Motivated by these findings, the authors ask whether any rule can perform better than F in the same 3×3 Moore neighborhood class and state that they suspect F is optimal but lack a proof, leaving the question open.