  The first year of JWST has revealed a surprisingly large number of luminous
galaxy candidates beyond $z>10$. While some galaxies have already been
spectroscopically confirmed, there is mounting evidence that a subsample of the
candidates with particularly red inferred UV colours are, in fact, lower
redshift contaminants. These interlopers are often found to be `HST-dark' or
`optically faint' galaxies at $z\sim2-6$, a population that is key to improving
our understanding of dust-obscured star formation throughout cosmic time. This
paper demonstrates the complementarity of ground-based mm-interferometry and
JWST infrared imaging to unveil the true nature of red 1.5-2.0 $\mu \rm{m}$
dropouts that have been selected as ultra-high-redshift galaxy candidates. We
present NOEMA Polyfix follow-up observations of four JWST red 1.5-2.0 $\mu
\rm{m}$ dropouts selected by Yan et al., 2023 as ultra-high-redshift candidates
in the PEARLS-IDF field. The new NOEMA observations constrain the rest-frame
far-infrared continuum emission and efficiently discriminate between
intermediate- and high-redshift solutions. We report $>10\sigma$ NOEMA
continuum detections of all our target galaxies at observed frequencies of $\nu
= 236$ and $252\ \rm{GHz}$, with FIR slopes indicating a redshift of $z<5$. We
modelled their optical-to-FIR spectral energy distribution (SED) with multiple
SED codes, finding that they are not $z>10$ galaxies but dust-obscured, massive
star-forming galaxies at $z\sim 2-4$ instead. The contribution to the cosmic
star formation rate density (CSFRD) of such sources is not negligible at
$z\simeq 3.5$ ($\phi\gtrsim(1.9-4.4)\times10^{-3}\ \rm{cMpc}^{-3}$; or $>3-6\%$
of the total CSFRD), in line with previous studies of optically faint and
sub-millimeter galaxies. This approach opens up a new window onto obscured star
formation at intermediate redshifts [abridged].
