Measuring the 2S-2P Lamb shift in a hydrogen-like muonic atom allows one to extract its nuclear charge radius with a high precision that is limited by the uncertainty in the nuclear structure corrections. The charge radius of the proton thus extracted was found to be 7-sigma away from the CODATA value, in what has become the yet unsolved proton radius puzzle. Further experiments currently aim at the isotopes of hydrogen and helium: the precise extraction of their radii may provide a hint at the solution of the puzzle. We present the first ab initio calculation of nuclear structure corrections, including the nuclear polarization correction, to the 2S-2P transition in $mu^3$He$^+$ and $mu^3$H, and assess solid theoretical error bars. Our predictions reduce the uncertainty in the nuclear structure corrections to the level of a few percents and will be instrumental to the on-going $mu^3$He$^+$ experiment. We also support the mirror $mu,^3$H system as a candidate for further probing of the nucleon polarizabilities and shedding more light on the puzzle.
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