Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag-Solitar groups

We say that a group has property $R_{infty}$ if any group automorphism has an infinite number of twisted conjugacy classes. Felshtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property $R_{infty}$. We define a solvable generalization $Gamma(S)$ of these groups which we show to have property $R_{infty}$. We then show that property $R_{infty}$ is geometric for these groups, that is, any group quasi-isometric to $Gamma(S)$ has property $R_{infty}$ as well.