Wightman QFTs corresponding to Minimal Models
Per suggestion, I am cross-posting the following question from MathOverflow (original). At MO, <PERSON> referenced the work [2]. Glancing through it, is not immediately obvious to me how far the work therein gets us along a mathematical construction (in the spirit of the Glimm-Jaffe school) of say the <PERSON> QFT corresponding to $\mathcal{M}(4,3)$. So as an addendum to the original question below: (1) how far does it get us? (2) concretely, how could one use the ideas of [2] to prove that the $S_N$ below satisfy reflection positivity?
Is it known (rigorously) whether or not there exist (1+1)D Wightman QFTs which can (in some reasonable sense) be said to correspond to physicists' unitary minimal models $\mathcal{M}(m+1,m)$, $m\in\mathbb{N}^+$, $m\geq 3$?
For example, the case $m=3$ is supposed to model the continuum 2D Ising model at criticality [1], and it seems that based on some heuristics physicists have conjectured -- or proven, at a level of rigor acceptable to them -- that the <PERSON> functions of such a QFT should be $$ S_N(z_1,\cdots,z_N) = \Big[\sum_{\substack{\varsigma_1,\ldots,\varsigma_N \in {-1,+1}\ \varsigma_1+\cdots+\varsigma_N=0}} \prod_{1\leq m<n\leq N} |z_m-z_n|^{\varsigma_m\varsigma_n/2} \Big]^{1/2},$$ $z_1,\ldots,z_N\in \mathbb{R}^2$ pairwise distinct. See Section 12.3.3 of [1]. 324
(Moreover, some aspects of this have been rigorously proven. See the answer on this question.) Since a Wightman QFT is determined by its <PERSON> functions, and since its <PERSON> functions are determined via analytic continuation by its <PERSON> functions, I would expect that the sequence ${S_N}_{N=0}^\infty$ uniquely specifies a Wightman QFT. Since one can analytically continue $S_N$ back to ``real time'' by hand, I would further expect that to construct such a QFT it suffices to check that the $S_N$ satisfy the original Osterwalder-Schrader axioms, of which the trickiest seems to be reflection positivity (/Wightman positivity after analytic continuation). Hence, I would expect that a necessary and sufficient condition for there to exist a Wightman QFT whose <PERSON> functions are the $S_N$ is that they satisfy reflection positivity, but this (if true) seems non-obvious to me.
[1] <PERSON>, <PERSON>, and <PERSON>, Conformal Field Theory, Chapters 7, 12
[2] <PERSON>, <PERSON>, <PERSON>, Distributions in CFT: Part II. Minkowski Space 346