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tokens: assumptions on the mass profile. By introducing a mass-sheet degeneracy parameter, we add the maximal degree of freedom in H 0 while having minimal constraining power by lensing data on their own. This is the most conservative approach when adding a single degree of freedom in our analysis. While mathematically this result is clearly understood, it is worth discussing the physical interpretation of this choice. If we had perfect cosmological numerical simulations or perfect knowledge of the internal mass distribution within elliptical galaxies, we would not have to worry about the internal MST. The approach chosen by our collaboration ( Wong et al. 2020 ; Shajib et al. 2019 ; Millon et al. 2020 ) was to assume physically motivated mass profiles with degrees of freedom in their parameters. In particular, the collaboration used two different mass profiles, a power-law elliptical mass profile, and a composite mass profile separating the luminous component (with fixed mass-to-light ratio) and a dark component described as a NFW profile. The good fit to the data, the small pixellated corrections on the profiles from the first lens system ( Suyu et al. 2010 ), and the good agreement of H 0 inferred with the two mass profiles was a positive sanity check on the result ( Millon et al. 2020 ). In this paper we have taken a different viewpoint, and asked how much can the mass profiles depart from a power-law and still be consistent with the data. By phrasing the question in terms of the MST we can conveniently carry out the calculations, because the MST leaves the lensing observables unchanged and therefore it corresponds to minimal constraints and assumptions, and thus maximal uncertainties with one additional degree of freedom. However, after the inference, one has to examine the inferred MST transformed profile and evaluate it in comparison with existing and future data to make sure it is realistic. We know that the exact MST cannot be the actual answer because profiles have to go zero density at large radii, but the approximate MST discussed in Sect. 2 provides a convenient interpretation with the addition of a cored mass component. Figure 17 illustrates a cored mass component approximating the MST inferred from this work, λ int = 0.91 ± 0.04, in combination with a power-law model inferred from the population mean of the SLACS analysis by Shajib et al. (2020b) . The analysis presented here guarantees that the inferred mass profile is consistent with the properties of TDCOSMO and SLACS lenses. We discuss below how additional data may allow us to constrain the models even further and thus reduce the overall uncertainty while keeping the assumptions at a minimum. Fig. 17. Illustration of the inferred mass profile of the joint TDCOSMO+SLACS SDSS + IFU analysis. A pure power-law with γ pl = 2.10 ± 0.05 is shown in orange. In blue is the result of this work of λ int = 0.91 ± 0.045 when interpreted as a cored mass component with R c uniform in [3′′,