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We derive the hydrodynamic limit of the Kawasaki dynamics for the one-dimensional conservative system of unbounded real-valued spins with arbitrary strong, quadratic and finite-range interactions. This significantly extends prior results for bounded interaction by Rezakhanlou and complements results obtained by H.T. Yau. The result is obtained by adapting two-scale approach of Grunewald, Otto, Villani and Westdickenberg combined with the authors’ recent approach on conservative systems with strong interactions.