The balanced-presentation conjecture for groups with 4-periodic cohomology

Let $G$ be a group with $4$-periodic cohomology. Balanced-presentation conjecture. Then $G$ has a balanced presentation. This would imply that groups with $4$-periodic cohomology are not counterexamples to the D2 problem; the paper notes that balanced presentations are not known for some examples, including $P_{48 \cdot 3}”$ and $Q(16;3,5)$.