Non-inner amenability conjecture for Röver–Nekrashevych groups

Let $G$ be a self-similar group of automorphisms of the infinite rooted $n$-ary tree, and let $V_n(G)$ be the Röver–Nekrashevych group obtained by combining $V_n$ with $G$. Non-inner amenability conjecture. The groups $V_n(G)$ are all non-inner amenable. This would extend the paper's main result from Higman–Thompson groups to Röver–Nekrashevych groups; understanding centralizers in $V_n(G)$ appears to be the main obstacle, and the conjecture remains open.