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First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\\mathcal E^p$ is a $d_p$-lsc functional that is convex along finite energy geodesics. Second, following the program of J. Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space $(\\mathcal E^2,d_2)$. 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