All the reviewers and the meta-reviewers are for accepting the paper, given that a decent job is put into rewriting it as indicated in the (updated) reviews. Indeed, the clarity is the main issue with this submission, and we except the authors to address it seriously thanks to the very thorough (all expert) reviews, as they indicate in their response and to their message to the AC. The request of the authors about low score of R2 and the length of the proofs (...) was also taken into the consideration when making the decision.