*Cosmo Comments*. The following comments can also be viewed as annotations on the paper via Hypothesis.

This paper studies the holographic dark energy (HDE) model, inspired by the proposal that in a given region of the Universe the infrared cutoff and the ultraviolet cutoff should be related. In particular, the maximum total energy in a region of the Universe set by the ultraviolet cutoff should saturate the mass of a black hole with size given by this region. If this size is taken to be the future event horizon of the Universe, one recovers an effective energy component which could drive cosmic acceleration.

The main result of this paper is that, after a fit to a combination of Planck and late-time datasets (BAO, SNe from Pantheon, and a prior on the local value of $H_0$ from Riess et al. 2019 [hereafter R19]), the HDE model can potentially solve the $H_0$ tension, yielding a best-fit $H_0=72.06$ km/s/Mpc from the Planck+BAO+Pantheon+R19 dataset combination. This result is potentially interesting because, if correct, it would by-pass the "no-go theorem" which argues against the impossibility of constructing a late-time solution to the $H_0$ tension, based on quasi-model-independent fits to the BAO+cosmographic SNe inverse distance ladder (in this sense, the BAO+SNe 2D $H_0$-$r_\mathrm{drag}$ contours in Fig. 1 from PRD 101 (2020) 043533 by Knox & Millea is emblematic).

I have three questions about how the analysis has been performed by the authors.

- How did the authors treat perturbations in the HDE fluid? Did they just use the standard Boltzmann equations found e.g. in Ma & Bertschinger for a fluid with equation of state $w(z)$ given by Eq. (4)? In other words, are the Boltzmann equations for HDE the same as those for a standard $w(z)$CDM model (which could be e.g. quintessence)?
- The authors choose to only use the $z>0.2$ SNe from Pantheon, because SNe at lower redshift prefer a lower $H_0$ which contradicts R19. However, this is
**exactly**the reason why one should use all SNe to see if the $H_0$ tension can be truly solved with all available data. Cutting out the $z<0.2$ SNe just because they give a worse fit to the full dataset combination including R19 is tantamount to cherry-picking one's data. I would recommend not cutting out the low-redshift SNe. Similarly, in the main text, at least as far as I saw, the authors did not discuss the results for the full dataset combination Planck+BAO+Pantheon+R19. They only quote the best-fit $H_0$ in Tab. I, which is $H_0=72.06$ km/s/Mpc. However, rather than its best-fit value, a more illuminating quantity would be the 68% CL interval on $H_0$ read off from its full posterior. I would recommend the authors add this number to the discussion. After all, they did quote this quantity for the Planck+BAO+R19 combination. On a side note, I think quoting this number in the abstract is a bit misleading, and the number that really should be quoted is the one from the Planck+BAO+Pantheon**or**Planck+BAO+Pantheon+R19 dataset combination. - Adding the R19 prior in all dataset combinations is a risky business. It makes sense to use the R19 prior only if the model at hand solves the $H_0$ tension
**before adding such a prior**. As a rule of thumb, the Planck+BAO+Pantheon and R19 contours for $H_0$ should overlap within about 2 sigma for the full combination to be meaningful. However, the authors never discuss the Planck+BAO+Pantheon dataset combination alone. Therefore, I do not know whether we can trust the Planck+BAO+Pantheon+R19 dataset combination. It would be illuminating to discuss the Planck+BAO+Pantheon dataset combination, including the 68% CL interval on $H_0$, both in the main text and in the abstract.

*[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]*