  We explore the potential of using the low-redshift Lyman-$\alpha$
(Ly$\alpha$) forest surrounding luminous red galaxies (LRGs) as a tool to
constrain active galactic nuclei (AGN) feedback models. Our analysis is based
on snapshots from the Illustris and IllustrisTNG simulations at a redshift of
$z=0.1$. These simulations offer an ideal platform for studying the influence
of AGN feedback on the gas surrounding galaxies, as they share the same initial
conditions and underlying code but incorporate different feedback
prescriptions. Both simulations show significant impacts of feedback on the
temperature and density of the gas around massive halos. Following our previous
work, we adjusted the UV background in both simulations to align with the
observed number density of Ly$\alpha$ lines ($\rm dN/dz$) in the intergalactic
medium and study the Ly$\alpha$ forest around massive halos hosting LRGs, at
impact parameters ($r_{\perp}$) ranging from 0.1 to 100 pMpc. Our findings
reveal that $\rm dN/dz$, as a function of $r_{\perp}$, is approximately 1.5 to
2 times higher in IllustrisTNG compared to Illustris up to $r_{\perp}$ of $\sim
10$ pMpc. To further assess whether existing data can effectively discern these
differences, we search for archival data containing spectra of background
quasars probing foreground LRGs. Through a feasibility analysis based on this
data, we demonstrate that ${\rm dN/dz} (r_{\perp})$ measurements can
distinguish between feedback models of IllustrisTNG and Illustris with a
precision exceeding 12$\sigma$. This underscores the potential of ${\rm dN/dz}
(r_{\perp})$ measurements around LRGs as a valuable benchmark observation for
discriminating between different feedback models.
