8 problems

Q36 presentation conjecture. If $\mathcal{P}=\mathcal{P}_9(n_1,n_2;m_1,m_2)$ presents $Q_{36}$, then

Exotic-presentation conjecture. The existence of an exotic presentation for $Q_{4n}$ should hold unconditionally, without assuming that $Q_{4n}$ has the D2 property.

D2 and presentation-form conjecture. If $n \ge 2$, then $Q_{4n}$ has the D2 property and every balanced presentation is homotopy equivalent to a presentation of the form…

Let $p$ be a prime satisfying $p\equiv1$ (mod 6). Let $\mathcal{Q}_{p}[0]$ be the quaternionic subgroup of level $0$, let $\mathcal{K}_{p}$ be the relevant free group, let $\mu_p$…

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 $IQ_{4n}^*$.

Let a quaternion brace be a brace whose multiplicative group is isomorphic to a generalized quaternion group. Power-of-two quaternion-brace conjecture. There are seven isomorphism…

For $m\in\mathbb{N}$, let $Q_{4m}$ be the generalized quaternion group, and call a brace a quaternion brace if its multiplicative group is isomorphic to a generalized quaternion gr…