Exponential decay of correlations in the 2D O(N) model at all temperatures

Classical $O(N)$ model on $Z^{2}$, $N \ge 3$, spins $s_{x}$ in $S^{N-1}$, Gibbs weight exp(beta sum_<xy> s_x . s_y). Prove that for every $\beta < \infty$ there are $C, m(\beta) > 0$ with $\langle s_{0} . s_{x}\rangle \le C\ \exp(-m(\beta)\ \mid x\mid)$ uniformly in the volume. For $N = 2$ the opposite holds at large $\beta$ (power-law decay, Frohlich and Spencer, 1981). Answer: a proof, or a proof of a massless low-temperature phase.

A proof of a positive mass $m(\beta)$ for all finite $\beta$, or a proof of power-law decay at some finite $\beta$, for $N = 3$.