Dynamical Tits alternative for groups of almost automorphisms of trees

We prove a dynamical variant of the Tits alternative for the group of almost automorphisms of a locally finite tree $\mathcal{T}$: a group of almost automorphisms of $\mathcal{T}$ either contains a nonabelian free group playing ping-pong on the boundary $\partial \mathcal{T}$, or the action of the group on $\partial \mathcal{T}$ preserves a probability measure. This generalises to all groups of tree almost automorphisms a result of S. Hurtado and E. Militon for Thompson's group $V$, with a hopefully simpler proof.