3 problems

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 a…

Presentation-form conjecture. Every such $Y$ is homotopy equivalent to a presentation of $Q_{4n}$ of the form

Second homotopy module conjecture. No finite presentation of $Q_{4n}$ has a second homotopy group other than $IQ_{4n}^*$.