  Black hole X-ray binaries in the quiescent state (Eddington ratios typically
$\lesssim$10$^{-5}$) display softer X-ray spectra (photon indices
$\Gamma\sim2$) compared to higher-luminosity black hole X-ray binaries in the
hard state ($\Gamma\sim1.7$). However, the cause of this softening, and its
implications for the underlying accretion flow, are still uncertain. Here, we
present quasi-simultaneous X-ray and radio spectral monitoring of the black
hole X-ray binary MAXI J1820$+$070 during the decay of its 2018 outburst and of
a subsequent re-flare in 2019, providing an opportunity to monitor a black hole
X-ray binary as it actively transitions into quiescence. We probe 1-10 keV
X-ray luminosities as low as $L_{\rm X}\sim4\times10^{32}$ erg s$^{-1}$,
equivalent to Eddington fractions of $\sim4\times10^{-7}$. During its decay
towards quiescence, the X-ray spectrum of MAXI J1820$+$070 softens from
$\Gamma\sim1.7$ to $\Gamma\sim2$, with the softening taking $\sim30$d, and
completing at $L_{\rm X}\approx10^{34}$ erg s$^{-1}$ ($\approx10^{-5} L_{\rm
Edd}$). While the X-ray spectrum softens, the radio spectrum generally remains
flat/inverted throughout the decay. We also find that MAXI J1820$+$070 follows
a radio ($L_{\rm R}$) -- X-ray luminosity correlation of the form $L_{\rm R}
\propto L_{\rm X}^{0.52\pm0.07}$, making it the fourth black hole system to
follow the so-called `standard track' unbroken over several (in this case,
four) decades in $L_{\rm X}$. Comparing the radio/X-ray spectral evolution(s)
with the $L_{\rm R}$ -- $L_{\rm X}$ plane, we find that the X-ray softening is
consistent with X-rays produced by Comptonization processes in a radiatively
inefficient accretion flow. We generally disfavor X-ray emission originating
solely from within the jet, with the possible exception of X-rays produced via
synchrotron self-Compton processes.
