Scaling limit of the subdiffusive random walk on a Galton-Watson tree in random environment

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable L'evy process, jointly with the convergence of the renormalised range of the walk towards the real tree coded by the latter continuous-time height process.