Lipschitz constants to curve complexes for punctured surfaces

Given a surface Sg,n there is a map sys:Tg,n→Cg,n where Tg,n is the Teichmüller space with the Teichmüller metric, Cg,n is the curve complex with the standard metric, anddCg,n(sys(X),sys(Y))≤KdTg,n(X,Y)+C. We give asymptotic bounds for the minimal value of K which we denote Kg,n≍1log⁡(|χg,n|) for sequences of surfaces with fixed genus and sequences of surfaces where the genus is a rational multiple of the punctures. This generalizes work of Gadre, Hironaka, Kent, and Leininger where they give the same asymptotic bounds for closed surfaces.