Stable trees as mixings of inhomogeneous continuum random trees

It has been claimed in Aldous, Miermont and Pitman that all Lévy trees are mixings of inhomogeneous continuum random trees. We give a rigorous proof of this claim in the case of a stable branching mechanism, relying on a new procedure for recovering the tree distance from the graphical spanning trees that works simultaneously for stable trees and inhomogeneous continuum random trees.