Rosenberger's Tits alternative conjecture for generalised triangle groups

Let $p,q,r>1$, and let $W$ be a cyclically reduced word of free-product length $2k\geq 2$ in $\mathbb{Z}_p*\mathbb{Z}_q$. Define the generalised triangle group

$$G=\langle x,y\mid x^p=y^q=W(x,y)^r=1\rangle.$$

Rosenberger's conjecture. If $G$ is a generalised triangle group, then either $G$ is virtually solvable or $G$ contains a non-abelian free subgroup.

This extends the Tits alternative from ordinary triangle groups to generalised triangle groups. The conjecture is proved in several cases, including $p=q=2$, $k\leq 4$, $r>2$, $q>5$, and $p>2$, but remains open in the case studied in the paper, namely $(p,q,r)=(2,4,2)$.