  A search for $CP$ violation in charmless four-body decays of $\Lambda_b$ and
$\Xi_b^0$ baryons with a proton and three charged mesons in the final state is
performed. To cancel out production and detection charge-asymmetry effects, the
search is carried out by measuring the difference between the $CP$ asymmetries
in a charmless decay and in a decay with an intermediate charmed baryon with
the same particles in the final state. The data sample used was recorded in
2011 and 2012 with the LHCb detector and corresponds to an integrated
luminosity of $3 {\rm fb}^{-1}$. A total of 18 $CP$ asymmetries are considered,
either accounting for the full phase space of the decays or exploring specific
regions of the decay kinematics. No significant $CP$-violation effect is
observed in any of the measurements.
