  We compute the baryonic screening masses with nucleon quantum numbers and its
negative parity partner in thermal QCD with $N_f=3$ massless quarks for a wide
range of temperatures, from $T \sim 1$ GeV up to $\sim 160$ GeV. The
computation is performed by Monte Carlo simulations of lattice QCD with
$O(a)$-improved Wilson fermions by exploiting a recently proposed strategy to
study non-perturbatively QCD at very high temperature. Very large spatial
extensions are considered in order to have negligible finite volume effects.
For each temperature we have simulated $3$ or $4$ values of the lattice
spacing, so as to perform the continuum limit extrapolation with confidence at
a few permille accuracy. The degeneracy of the positive and negative
parity-state screening masses, expected from Ward identities associated to
non-singlet axial transformations, provides further evidence for the
restoration of chiral symmetry in the high temperature regime of QCD. In the
entire range of temperatures explored, the baryonic masses deviate from the
free theory result, $3 \pi T$, by $4$-$8\%$. The contribution due to the
interactions is clearly visible up to the highest temperature considered, and
cannot be explained by the expected leading behavior in the QCD coupling
constant $g$ over the entire range of temperatures explored.
