I'm trying to study a more mathematically rigorous approach to Conformal Field Theory in two-dimensions. In particular I'm looking for a clear description of what we should take as axioms defining a CFT in contrast to what can be understood as consequence of those axioms. In fact, Physics texts are more often quite confusing in that regard because they do not give a clear distinction between what they are assuming and what they are proving, or what they are just claiming without no proof whatsoever.
In that regard, I started studying the book "A Mathematical Introduction to Conformal Field Theory" by Schottenloher. The issue, though, is that I have found what seems to be a major mistake in the book. As discussed in this Phys.SE thread, in proposition (9.8) the author claims that the Conformal Ward Identities are $$0 = \sum_{j=1}^n (z_j^{m+1} \partial_{z_j} + (m+1) h_j z_j^m ) \langle \phi_1(z_1) \ldots \phi_j(z_n) \rangle,$$ which is non-sequitur: the LHS is not zero for $|m|>1$.
This got me quite worried with the book, because then I'm not sure I can trust what he is saying to be results that follow from the axioms he presented.
So, what are other books or notes that like Schottenloher, present a mathematically rigorous approach to Conformal Field Theory in two-dimensions, giving axioms to define such theories together with what are actually theorems proved from these axioms? Moreover, apart from this mistake with the Conformal Ward Identities, is Schottenloher's book trustworthy in its remaining contents?
Although a bit old, the "Lectures on Conformal Field Theory" by K. Gawedzski would be a good place to start, in order to get an overall picture.
There has been also recent progress on rigorously justifying the machinery of CFT in the case of Liouville theory. See in particular:
"Conformal bootstrap in Liouville Theory" by Guillarmou, Kupiainen, Rhodes, and Vargas,
and
"Segal's axioms and bootstrap for Liouville Theory" by the same authors.