We plan to measure several 2S–2P transition frequencies in μ⁴He⁺ and μ³He⁺ by means of laser spectroscopy with an accuracy of 50 ppm. This will lead to a determination of the corresponding nuclear rms charge radii with a relative accuracy of 3 × 10⁻⁴, limited by the uncertainty of the nuclear polarization contribution. First, these measurements will help to solve the proton radius puzzle. Second, these very precise nuclear radii are benchmarks for ab initio few-nucleon theories and potentials. Finally when combined with an ongoing measurement of the 1S–2S transition in He⁺, these measurements will lead to an enhanced bound-state QED test of the 1S Lamb shift in He⁺.