Scaling limit of the random walk on a Galton-Watson tree with regular varying offspring distribution

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order $\kappa \in (1,2)$. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the continuum tree coded by the latter continuous-time height process.