Generalised triangle groups of type $\mathbf{(3,q,2)}$

If $G$ is a group with a presentation of the form $\langle x,y|x^3=y^q=W(x,y)^2=1\rangle$, then either $G$ is virtually soluble or $G$ contains a free subgroup of rank $2$. This provides additional evidence in favour of a conjecture of Rosenberger.