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The STAR experiment reports new, high-precision measurements of the transverse single-spin asymmetries for π^{±} within jets, namely the Collins asymmetries, from transversely polarized p^{↑}p collisions at sqrt[s]=510 GeV. The energy-scaled distribution of jet transverse momentum, x_{T}=2p_{T,jet}/sqrt[s], shows a remarkable consistency for Collins asymmetries of π^{±} in jets between sqrt[s]=200 GeV and 510 GeV. This indicates that the Collins asymmetries are nearly energy independent, with, at most, a very weak scale dependence in p^{↑}p collisions. These results extend to high-momentum scales (Q^{2}≤3400 GeV^{2}) and enable unique tests of evolution and universality in the transverse-momentum-dependent formalism, thus providing important constraints for the Collins fragmentation functions.