Consider independent fair coin flips at each site of the lattice ℤ
. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤ
and is given by a deterministic function of the coin flips. Let Z
Φ be the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour of Z
Φ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZ
2/3-ε < ∞ for every ε > 0.