Construct the continuum $2\mathrm{D}$ $O(N)$ sigma model as an interacting theory
In plain words
Zooming out at the scale of the correlation length while the temperature goes to zero should give a continuum quantum field theory with massive interacting particles. Building it rigorously would be a two-dimensional model of the Yang-Mills construction problem.
Precise statement
Show that lattice $O(N)$ correlations ($N \ge 3$) at separations $x = y\,\xi(\beta),\ y\ \text{fixed}$, converge as $\beta \to \infty$ to the Schwinger functions of a non-Gaussian Euclidean field theory on $R^2$ satisfying the Osterwalder-Schrader axioms with a mass gap, and identify its scattering with the factorized S-matrix of A. B. and Al. B. Zamolodchikov (1979). Answer: a construction with proof.
What would settle it
A proof of convergence of the rescaled lattice correlations to an Osterwalder-Schrader theory with non-Gaussian four-point function.