The Pierre Auger Observatory (Auger) and the Telescope Array (TA), located, respectively, in
the Southern and Northern hemispheres, are the largest ultra-high-energy cosmic ray (UHECR)
observatories. The Auger and TA Collaborations have collected unprecedented statistics providing
us with a unique opportunity to search for the differences between the UHECR energy spectra and
mass compositions in the complementary sky regions. To correctly attribute such differences to
the properties of the UHECR sources or propagation, the systematic effects in the measurements
of each observatory should be considered properly. In this context, the task of the Auger – TA
mass composition working group is to identify possible differences of astrophysical origin in
the measurements of the depth of the maximum of air-shower profiles, $X_{\rm max}$, performed at both
observatories using the fluorescence technique. Due to distinct approaches to event selection and
analysis at Auger and TA, the working group uses a specially designed method to transfer the Auger
$X_{\rm max}$ distributions into the TA detector. To this end, dedicated air-shower and detector simulations
for the TA Black Rock Mesa and Long Ridge fluorescence detector stations were performed with
the Sibyll 2.3d hadronic interaction model. From the comparison of the first two moments and the
shapes of $X_{\rm max}$ distributions for energies above $10^{18.2}$ eV, no significant differences between the
Auger and TA measurements were found.