The presentation-form conjecture for D(2)-complexes with quaternion fundamental group

Let $Q_{4n}$ be a quaternion group with $n\geq 2$, and let $Y$ be a $D(2)$--complex with

$$\pi_1(Y)=Q_{4n},\qquad \chi(Y)=1.$$

For elements or words $a,b$, let $\operatorname{Eq}(a,b)$ be an equation implied by $ab=1$ that equates words in $a,b$.

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

$$\mathcal{Q}=\langle x,y\mid y^2=x^n,\quad \operatorname{Eq}(y^{-1}xy,x)\rangle.$$

This would classify all minimal-Euler-characteristic $D(2)$--complexes with quaternion fundamental group by a specified family of presentations. The source says that this remains possible, while noting that the $D(2)$ property has been verified for $Q_{28}$; no resolution of this broader assertion is given.