  The stellar mass assembly of brightest cluster galaxies (BCGs) is still
debated. We have observed in CO with the IRAM-30m two star forming BCGs
belonging to SpARCS clusters, 3C 244.1 ($z=0.4$) and SDSS J161112.65+550823.5
($z=0.9$), and compared their molecular gas and star formation properties with
those of a compilation of $\sim100$ distant cluster galaxies, including nine
additional distant BCGs at ${z\sim0.4-3.5}$. We have set robust upper limits of
$M_{{\rm H}_2}<1.0\times10^{10}~M_\odot$ and $<2.8\times10^{10}~M_\odot$ to
their molecular gas content, respectively, as well as to the molecular gas to
stellar mass ratio $M({\rm H}_2)/M_\star\lesssim0.2$ and depletion time
$\tau_{\rm dep}\lesssim40$ Myr of the two targeted BCGs. They are thus among
the distant cluster galaxies with the lowest gas fractions and shortest
depletion times. The majority, $64\%\pm15\%$ and $73\%\pm18\%$, of the 11 BCGs
with observations in CO have lower $M({\rm H}_2)/M_\star$ and $\tau_{\rm dep}$,
respectively, than those estimated for main sequence galaxies. Statistical
analysis tentatively suggests that the values of $M({\rm H}_2)/M_\star$ and
$\tau_{\rm dep}$ for the 11 BCGs deviates, with a significance of
$\sim2\sigma$, from those of the comparison sample of cluster galaxies. A
morphological analysis for a subsample of seven BCGs with archival HST
observations reveals that $71\%\pm17\%$ of the BCGs are compact or show
star-forming substructures/components. Our results suggest a scenario where
distant star forming BCGs assemble a significant fraction $\sim16\%$ of their
stellar mass in the relatively short timescale $\tau_{\rm dep}$, while
environmental mechanisms might prevent the replenishment of gas feeding the
star formation. We speculate that compact components favor the rapid exhaustion
of molecular gas and help to quench the BCGs. Distant star forming BCGs are
excellent targets for ALMA and JWST.
