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when I look for it in Wikipedia I find that there are (mainly) two kinds of natural/geometrized units: The Planck ones and the Stoney ones,

(https://en.wikipedia.org/wiki/Geometrized_unit_system)

Also, some variations of the CGS system and Heaviside-Lorentz unit system share some of the desired magnitudes normalized to one

(https://en.wikipedia.org/wiki/Centimetre–gram–second_system_of_units#Various_extensions_of_the_CGS_system_to_electromagnetism)

(https://en.wikipedia.org/wiki/Lorentz–Heaviside_units)

but none of these systems seem to be compatible with ##\hbar=G=c=\mu_0=1## so...

The question is: Does exist a unit system with those specific magnitudes set to one?