I would like to know if there is an analogue of Culler & Vogtmann's outer space to study outer automorphisms of free pro-$p$ groups. Perhaps an initial guess of such a space would be a moduli space of minimal free actions of a fixed free pro-$p$ group on pro-$p$ trees, however I am not sure if this would have all the nice properties (contractibility, finite point stabilizers) enjoyed in the classical setting.
Many thanks!