The Tits alternative for generalized triangle groups of type $(3,4,2)$

A generalized triangle group is a group that can be presented in the form $G=\langle{x,y}|x^p=y^q=w(x,y)^r=1\rangle$ where $p,q,r\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least $2$ in the free product $\mathbb Z_p*\mathbb Z_q=\langle{x,y}{x^p=y^q=1}\rangle$. Rosenberger has conjectured that every generalized triangle group $G$ satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple $(p,q,r)$ is one of $(2,3,2)$, $(2,4,2)$, $(2,5,2)$, $(3,3,2)$, $(3,4,2)$ or $(3,5,2)$. Building on a result of Benyash–Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$.