Two infinite families of facets of the holographic entropy cone

1) This paper proves an interesting result in full technical detail.
2) The paper makes good decisions about which portions of the proof to relegate to the appendices, and which lemmas are important to the understanding of the proof and thus need to be in the main text.
3) The paper nicely puts their result in the context of the larger program to better understand and describe the holographic entropy cone.
4) The proof is mathematically solid, and its method provides a path for a possible interpretation of the toric family of inequalities in terms of 4- and 6- party entaglement.
5) The discussion nicely relates to the physics content of the inequalities at issue; the uniformly-weighted star graphs corresponded to old black holes with each leg corresponding to a previously emitted Hawking quantum. This reader will be interested to see further understanding along these lines.

All weaknesses are minor and easily remediable as per the requested changes below.

This paper proves that two infinite families of holographic entropy inequalities are as tight as possible. That is, for each inequality, they construct $2^N-2$ linearly independent saturating holographic entropy vectors, showing that the achievable saturating surface is codimension one in the $2^N-1$ dimensional entropy space.

The result is of interest first because it completes the mathematical understanding of this set of inequalities (by understanding when they can be saturated). It improves the understanding of the holographic cone, as some previous facets now have a generalization as facets. And most importantly, its relation to understanding the nature of old black holes, as well as to entanglement in four and six party arrangements, is likely to produce interesting results in the future.

As the abstract admits, the proof itself is technical, but the paper does a good job of presenting its main ideas in the body while relegating more technical details to the appendix. The techniques used in the proof are also relevant; as pointed out in the conclusion, the fact that only star-graphs are needed, and the interplay between the $K4$ and $K6$ vectors, may point to an interpretation of these inequalities in terms of 4 and 6 party entanglement.

1) Apparent typo in right hand side of equation 3.2, one argument reads $X_0^1, X_1^1, X_2^2,X_4^1$ but should read $X_0^1,X_1^1,X_2^1,X_4^1$.
2) Figure 5 in the appendix should say that $g(1)$ is marked in red while $g(2)$ is marked in green.
3) Occasional unusual phrasings , e.g. "How to decide on that?" or "No because otherwise".

Publish (easily meets expectations and criteria for this Journal; among top 50%)