  We present deep X-ray limits on the presence of a pre-explosion counterpart
to the low-luminosity Type Ia supernova (SN Ia) 2017ejb. SN 2017ejb was
discovered in NGC 4696, a well-studied elliptical galaxy in the Centaurus
cluster with 894 ks of Chandra imaging between 14 and 3 years before SN 2017ejb
was discovered. Using post-explosion photometry and spectroscopy of SN 2017ejb,
we demonstrate that SN 2017ejb is most consistent with low-luminosity SNe Ia
such as SN 1986G and SN 1991bg. Analyzing the location of SN 2017ejb in
pre-explosion images, we do not detect a pre-explosion X-ray source. We use
these data to place upper limits on the presence of any unobscured supersoft
X-ray source (SSS). SSS systems are known to consist of white dwarfs accreting
from a non-degenerate companion star. We rule out any source similar to known
SSS systems with $kT_{\rm eff} > 85$ eV and $L_{\rm bol} >
4\times10^{38}~\text{erg s}^{-1}$ as well as models of stably-accreting
Chandrasekhar-mass WDs with accretion rates
$\dot{M}>3\times10^{-8}~M_{\odot}~\text{yr}^{-1}$. These findings suggest that
low-luminosity SNe Ia similar to SN 2017ejb explode from WDs that are low-mass,
have low pre-explosion accretion rates, or accrete very soon before explosion.
Based on the limits from SN 2017ejb and other nearby SNe Ia, we infer that
$<$47% of SNe Ia explode in stably-accreting Chandrasekhar-mass SSS systems.
