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It is conjectured, that the parallel correlation function in the infinite volume Gibbs state, i.e., the map\n  $$\n  x \\mapsto \\langle \\cos\\theta(0) \\cos\\theta(x) \\rangle_{\\mu_{\\mathrm{Vil}, \\beta}} -\\left( \\langle \\cos\\theta(0) \\rangle_{\\mu_{\\mathrm{Vil}, \\beta}} \\right)^2,\n  $$\n  decays like $|x|^{-2(d-2)}$ as $|x| \\to \\infty$ at low temperature. The results of Bricmont, Fontaine, Lebowitz, Lieb, and Spencer (1981) show that for the related XY model, this correlation decays at least as fast as $|x|^{2-d}$. 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