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 Schwinger 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]. (Moreover, some aspects of this have been rigorously proven. See the answer on this question.) Since a Wightman QFT is determined by its Wightman functions, and since its Wightman functions are determined via analytic continuation by its Schwinger 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 Schwinger functions are the $S_N$ is that they satisfy reflection positivity, but this (if true) seems non-obvious to me.

[1] Francesco, Mathieu, and Senechal, *Conformal Field Theory*, Chapters 7, 12

modelsthat can be described by the axioms are massive and my recollecition is faulty $\endgroup$