  JWST has uncovered a substantial population of high-$z$ ($z \gtrsim 4$)
galaxies exhibiting broad H$\alpha$ emission line with a Full Width at Half
Maximum exceeding $1,000$~km~s$^{-1}$. This population includes a subset known
as 'Little Red Dots', characterized by their compact morphology and extremely
red rest-frame optical colors. If all of these broad H$\alpha$ emitters were
attributed to type $1-1.9$ Active Galactic Nuclei (AGNs), it would imply a
significantly higher number density of low-luminosity AGNs than extrapolated
from that of more luminous AGNs. Here, we have examined the rest-frame
ultraviolet (UV)-optical flux variability of five JWST broad H$\alpha$ emitters
using multi-epoch, multi-band JWST/NIRCam imaging data. The rest-frame temporal
sampling interval of the NIRCam data ($\sim 400-500$~days$/(1+z)$) is
comparable to typical variability timescales of AGNs with black hole (BH)
masses of $M_{\text{BH}} \sim 10^{7}~M_{\odot}$; thus, the flux variations
should be detectable if AGNs were present. However, no measurable flux
variation over the rest-frame wavelength range of $\lambda_{\text{rest}} \sim
1,500-9,000$\AA\ has been detected, placing stringent upper limits on the
variability amplitudes. This result, combined with the X-ray faintness
confirmed by the ultra-deep {\it Chandra} data, indicates that, under the AGN
scenario, we need to postulate peculiar Compton-thick broad-line AGNs with
either ($a$) an intrinsically non-variable AGN disk continuum, ($b$) a host
galaxy-dominated continuum, or ($c$) scattering-dominated AGN emission.
Alternatively, ($d$) they could be non-AGNs where the broad-line emission
originates from unusually fast and dense/low-metallicity star-formation-driven
outflows or inelastic Raman scattering of stellar UV continua by neutral
hydrogen atoms.
