  Recent {\em James Webb Space Telescope (JWST)} observations have revealed a
population of compact extragalactic objects at $z\gtrsim4$ with red
near-infrared colors, which have been dubbed as ``Little Red Dots" (LRDs). The
spectroscopically-selected LRDs exhibit broad H$\alpha$ emission lines, which
has led to the conclusion that type-I active galactic nuclei (AGN) are harbored
in the galaxies' dust-reddened cores. However, other mechanisms, like strong
outflowing winds, could also produce broad H$\alpha$ emission lines, and thus,
the nature of LRDs is still under debate. In this work, we test the AGN
hypothesis for LRDs by stacking the archival {\em Chandra} observations of 19
spectroscopically-selected LRDs. We obtain non-detections in both the soft
$(0.5-2\text{ keV})$ and the hard $(2-8\text{ keV})$ X-ray band after stacking,
and put upper limits on the X-ray luminosities of these LRDs. We find that the
soft band upper limit is $\sim1$ dex lower than the expected level from the
$L_\text{X}-L_{\text{H}\alpha}$ relation, while the hard band upper limit is
consistent with this relation. This result suggests that LRDs and typical
type-I AGNs have significantly different properties. We consider it unlikely
that absorption is the main reason for the observed low
$L_\text{X}/L_{\text{H}\alpha}$ ratios of LRDs. Alternatively, we discuss two
plausible hypotheses: (1) LRDs have intrinsically weak X-ray emissions, (2)
fast, galactic-scale outflows have a major contribution to the observed broad
H$\alpha$ lines. Our findings indicate that empirical relations (e.g., for
black hole mass measurements) established for typical type-I AGNs should be
used with caution when analyzing the properties of LRDs.
