The recently discovered Hosking integral governs the decay of non-helical
turbulence. There is a growing number of applications, for example to
neutron star crusts. Here, we study the Hosking integral in yet another
and completely different system, applicable to the early universe,
which includes the chiral chemical potential. The importance of our
paper, justifying the publication in PRR Letters, lies in promoting the
idea that the Hosking integral has broad applications. Particularly
important are those to cosmological MHD after axion inflation, which
can also be connected with the origin of the observed baryon asymmetry
of the Universe.