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[ref], with that obtained from the direct calculation; * comparison of the anti-symmetric part of the splitting function for $q →qq'q̅'$ with the result given in Ref. [ref] ; * comparison of the splitting functions for $q →qq'q̅'$ and $q →qqq̅$ expanded to $ϵ^0$ obtained using the methods of Section [ref], with the result of an expansion in the triple-collinear limit of one-loop matrix-elements squared for the processes $V →qq̅q'q̅'$ and $V →qq̅qq̅$, with $V$ a massive off-shell vector boson; * numerical comparison of the triple-collinear limits of one-loop matrix-elements squared at $ϵ^0$ for six-parton processes with the predicted asymptotics, Eq. 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