The quaternion commutator conjecture

Let $Q_{p,l}$ be the set of quaternions under consideration, let $x,y\in Q_{p,l}$, and write $\langle x,y\rangle'$ for the commutator subgroup of the subgroup generated by $x$ and $y$. Also write $Q_{p,l}'$ for the commutator subgroup associated with $Q_{p,l}$. Quaternion commutator conjecture.

$$-1\notin\langle x,y\rangle'.$$

Moreover, the stronger assertion is conjectured:

$$-1\notin Q_{p,l}'.$$

This conjecture would, together with the preceding lemma, imply that $Z\langle x,y\rangle\cap\langle x,y\rangle'=1$. The source provides no resolution.