The $R_\infty$-nilpotency degree of some lollipop GBS groups

We say that a group $G$ has the $R_{\infty}$ property if the number $R(φ)$ of twisted conjugacy classes is infinite for any automorphism $φ$ of $G$. The $R_{\infty}$-nilpotency degree is the least integer $c$ such that $G/γ_{c+1}(G)$ has property $R_{\infty}$; in case this integer does not exist, we