Resolving this would sharpen our understanding of which classical combinatorial theorems are truly choice-free and which secretly encode fragments of choice, contributing to the broader project of reverse mathematics in the set-theoretic context. It would also illuminate the structural theory of automorphism groups of infinite graphs in choiceless universes, with implications for algebra in ZF. More broadly, a clean independence result here would provide a new benchmark problem for calibrating set-theoretic principles, potentially connecting graph combinatorics to well-studied choice hierarchies such as those involving the ultrafilter lemma or Koenig's lemma.