Schematic view of the magnetized accretion-ejection structure as a function of radius r and altitude z. The extent of the JED and SAD is shown in gray (vertical heights were computed from our model, see Paper III), while the jet launched from the JED is shown in orange (sketch of what its geometry could look like). The photosphere r_ν(z) at a given frequency ν is shown in red, splitting the jet in an inner optically thick (dark orange) and outer optically thin (light orange) regions. Within our simple approach, each field line widens in a self-similar way so that at a given altitude z, a field line anchored at r_in has achieved a radius r(z) = Wr_in, while that anchored at r_J is Wr_J, with the same asymptotic widening factor W.