  The data on spectroscopic galaxy clustering collected by the Dark Energy
Spectroscopic Instrument (DESI) will allow the significant detection of subtle
features in the galaxy two-point correlation in redshift space, beyond the
"standard" redshift-space distortions. Here we present an independent
assessment of the detectability of the relativistic dipole in the
cross-correlation of two populations of galaxies if they would be selected from
the Bright Galaxy Survey (BGS) of DESI. We build synthetic galaxy catalogues
with the characteristics of the BGS using the light cone of a relativistic
$N$-body simulation. Exploring different ways of splitting the populations of
galaxies we find that with an unequal split with more bright galaxies than
faint galaxies the detectability is significantly boosted, reaching 19 $\sigma$
in the redshift bin $0.2 \lesssim z \lesssim 0.3$ and expected to be even
higher at lower redshift. Moreover, we find that the measured dipole agrees
very well with the prediction of relativistic effects from linear theory down
to separations of $\sim$ 30 Mpc/$h$.
