We compute flavor non-singlet meson screening masses in the chiral limit of QCD with $𝑁_{𝑓} = 3$ quarks. The calculation is carried out at 12 temperatures from 𝑇 ≈ 1 GeV up to the electroweak scale. For each temperature we simulated several lattice spacings, so as to be able to perform the continuum limit extrapolation with confidence at a few permille accuracy. In the entire range of temperatures explored, the meson screening masses deviate from the free theory result 2𝜋𝑇 by at most a few percent. Their values, however, cannot be explained by one-loop perturbation theory up to the electroweak scale, where the pseudoscalar and the vector screening masses are still significantly different within our precision. Chiral symmetry restoration manifests itself through the degeneracy of the pseudoscalar and the scalar channels and of the vector and the axial ones.