  Young brown dwarfs exhibit atmospheric characteristics similar to those of
super-Jupiters, providing a unique opportunity to study planetary atmospheres.
The ESO SupJup Survey, utilizing CRIRES$^+$ on the Very Large Telescope, aims
to assess the role of $^{12}$C/$^{13}$C as a formation tracer. We present
observations of three young brown dwarfs: 2MASS J12003792-7845082, TWA 28, and
2MASS J08561384-1342242, with the goal of constraining their chemical
compositions, thermal profiles, surface gravities, spin rotations, and
$^{12}$C/$^{13}$C. Atmospheric retrievals of CRIRES$^+$ K-band spectra were
conducted using the radiative transfer code petitRADTRANS coupled with the
Bayesian inference algorithm MultiNest, resulting in a detailed
characterization of the atmospheres of these objects. We report the volume
mixing ratios of main molecular and atomic species, including the novel
detection of hydrogen fluoride (HF) in a brown dwarf's atmosphere, and
determine $^{12}$C/$^{13}$C values of $81^{+28}_{-19}$ and $79^{+20}_{-14}$ in
the atmospheres of TWA 28 and J0856, respectively, with strong significance
($>3\sigma$). Tentative evidence ($\sim 2\sigma$) of $^{13}$C in J1200 was
found, with $^{12}$C/$^{13}$C = $114^{+69}_{-33}$, along with $^{18}$O detected
at moderate significance in J0856 (3.3$\sigma$) and TWA 28 (2.1$\sigma$). The
retrieved thermal profiles indicate hot atmospheres (2300-2600 K) with low
surface gravities and slow spins, consistent with young objects. The consistent
carbon isotope ratios among the three objects, showing no significant deviation
from the local ISM, suggest a fragmentation-based formation mechanism similar
to star formation. The tentative detection of $^{18}$O in two objects
highlights the potential of high-resolution spectroscopy to probe additional
isotope ratios, such as $^{16}$O/$^{18}$O, in the atmospheres of brown dwarfs
and super-Jupiters.
