ZF existence of a partition of R^3 into unit circles

Determine whether the Zermelo–Fraenkel axioms without the Axiom of Choice (ZF) suffice to prove the existence of a partition of the entire three-dimensional Euclidean space R^3 into circles of radius 1 that are pairwise disjoint and whose union is all of R^3.

Background

Partitions of R^3 into unit circles (PUCs) are known to exist in ZFC by classical choice-based constructions. Beyond choice, Szulkin showed R^3 can be partitioned into circles if one drops the equal-radius requirement, and recent work provides explicit partitions of unit circles for certain open subsets of R^3.

Despite these advances, the status of a full PUC in ZF alone remains unsettled. The paper constructs choiceless models where PUCs exist, but this does not resolve whether ZF itself proves their existence without additional assumptions.