  The central regions of nearby elliptical galaxies are dominated by baryons
(stars) and provide interesting laboratories for studying the radial
acceleration relation (RAR). We carry out exploratory analyses and discuss the
possibility of constraining the RAR in the super-critical acceleration range
$(10^{-9.5},\hspace{1ex}10^{-8})$~${\rm m}~{\rm s}^{-2}$ using a sample of
nearly round pure-bulge (spheroidal, dispersion-dominated) galaxies including
24 ATLAS$^{\rm 3D}$ galaxies and 4201 SDSS galaxies covering a wide range of
masses, sizes and luminosity density profiles. We consider a range of current
possibilities for the stellar mass-to-light ratio ($M_\star/L$), its gradient
and dark or phantom matter (DM/PM) halo profiles. We obtain the probability
density functions (PDFs) of the parameters of the considered models via
Bayesian inference based on spherical Jeans Monte Carlo modeling of the
observed velocity dispersions. We then constrain the DM/PM-to-baryon
acceleration ratio $a_{\rm X}/a_{\rm B}$ from the PDFs. Unless we ignore
observed radial gradients in $M_\star/L$, or assume unreasonably strong
gradients, marginalization over nuisance factors suggests $a_{\rm X}/a_{\rm B}
= 10^{p} (a_{\rm B}/a_{+1})^q$ with $p = -1.00 \pm 0.03$ (stat)
$^{+0.11}_{-0.06}$ (sys) and $q=-1.02 \pm 0.09$ (stat) $^{+0.16}_{-0.00}$ (sys)
around a super-critical acceleration $a_{+1}\equiv 1.2\times 10^{-9}~{\rm
m}~{\rm s}^{-2}$. In the context of the $\Lambda$CDM paradigm, this RAR
suggests that the NFW DM halo profile is a reasonable description of galactic
halos even after the processes of galaxy formation and evolution. In the
context of the MOND paradigm, this RAR favors the Simple interpolating function
but is inconsistent with the vast majority of other theoretical proposals and
fitting functions motivated mainly from sub-critical acceleration data.
