  We investigate the structural relaxation of a soft-sphere liquid quenched
isochorically ($\phi=0.7$) and instantaneously to different temperatures $T_f$
above and below the glass transition. For this, we combine extensive Brownian
dynamics simulations and theoretical calculations based on the non-equilibrium
self-consistent generalized Langevin equation (NE-SCGLE) theory. The response
of the liquid to a quench generally consists of a sub-linear increase of the
$\alpha$-relaxation time with system's age. Approaching the ideal
glass-transition temperature from above ($T_f>T^a$) sub-aging appears as a
transient process describing a broad equilibration crossover for quenches to
nearly arrested states. This allows us to empirically determine an
equilibration timescale $t^{eq}(T_f)$ that becomes increasingly longer as $T_f$
approaches $T^a$. For quenches inside the glass ($T_f\leq T^a$) the growth rate
of the structural relaxation time becomes progressively larger as $T_f$
decreases and, unlike the equilibration scenario, $\tau_{\alpha}$ remains
evolving within the whole observation time-window.These features are
consistently found in theory and simulations with remarkable semi-quantitative
agreement, and coincide with those revealed in the similar and complementary
exercise [Phys. Rev. {\bf 96}, 022608 (2017)] that considered a sequence of
quenches with fixed final temperature $T_f=0$ but increasing $\phi$ towards the
hard-sphere dynamical arrest volume fraction $\phi^a_{HS}=0.582$. The NE-SCGLE
analysis, however, unveils various fundamental aspects of the glass transition,
involving the abrupt passage from the ordinary equilibration scenario to the
persistent aging effects that are characteristic of glass-forming liquids. The
theory also explains that, within the time window of any experimental
observation, this can only be observed as a continuous crossover.
