  We combine the giant equilibration speedup provided by the swap Monte Carlo
algorithm to extensive multi-CPU molecular dynamics simulations to explore the
first ten decades of the equilibrium dynamics of supercooled liquids near the
experimental glass transition. We demonstrate in particular the emergence of a
power law in relaxation spectra, in quantitative agreement with the excess
wings observed experimentally. Detailed analysis of particle motion shows that
this power law directly originates from broadly distributed processes
experienced by rare, localised regions relaxing much before the bulk. We
explain why these wings appear in "excess" by constructing an empirical model
associating heterogeneous activated dynamics and dynamic facilitation, which
appear to be the two minimal physical ingredients revealed by our simulations.
