  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.
