Except for the part where discusses at length his criticisms with the paper itself, which apparently you don't wish to comment on.

That's unlikely to be the "source" of his skepticism - rather, his understanding of the theoretical foundations of the subject would be the source. That same understanding is the basis for his ability to identify possible flaws.

Not everyone is an armchair commentator merely repeating their unsubstantiated opinions. In Kalai's case, you could just say "Kalai's conjectures" and people familiar with the field are likely to recognize what you're referring to, although the may ask "which ones?" since he's produced several well-known ones in different math subfields.