  The discovery of Little Red Dots (LRDs) -- a population of compact,
high-redshift, dust-reddened galaxies -- is one of the most surprising results
from JWST. However, the nature of LRDs is still debated: does the near-infrared
emission originate from accreting supermassive black holes (SMBHs), or intense
star formation? In this work, we utilize ultra-deep Chandra observations and
study LRDs residing behind the lensing galaxy cluster, Abell~2744. We probe the
X-ray emission from individual galaxies but find that they remain undetected
and provide SMBH mass upper limits of
$\lesssim(1.5-16)\times10^{6}~\rm{M_{\odot}}$ assuming Eddington limited
accretion. To increase the signal-to-noise ratios, we conduct a stacking
analysis of the full sample with a total lensed exposure time of
$\approx87$~Ms. We also bin the galaxies based on their stellar mass, lensing
magnification, and detected broad-line H$\alpha$ emission. For the LRDs
exhibiting broad-line H$\alpha$ emission, there is a hint of a stacked signal
($\sim2.6\sigma$), corresponding to a SMBH mass of
$\sim3.2\times10^{6}~\rm{M_{\odot}}$. Assuming unobscured, Eddington-limited
accretion, this BH mass is at least 1.5 orders of magnitude lower than that
inferred from virial mass estimates using JWST spectra. Given galaxy-dominated
stellar mass estimates, our results imply that LRDs do not host over-massive
SMBHs and/or accrete at a few percent of their Eddington limit. However,
alternative stellar mass estimates may still support that LRDs host
over-massive BHs. The significant discrepancy between the JWST and Chandra data
hints that the scaling relations used to infer the SMBH mass from the H$\alpha$
line and virial relations may not be applicable for high-redshift LRDs.
