The conjugacy problem for Thompson-like groups

In this paper we generalize techniques of Belk-Matucci to solve the conjugacy problem for every Thompson-like group $V_n(H)$, where $n\ge 2$ and $H$ is a subgroup of the symmetric group on $n$ elements. We use this to prove that, if $n\mathrm{\ne }m$, $V_n(H)$ is not isomorphic to $V_m(G)$ for any $H,G$.