  We clarify and extend further the idea developed in \cite{PKLMR,MLPR} that
baryonic matter at high density has an emergent "pseudo-conformal symmetry." It
is argued that as baryonic density exceeds $n \simeq n_{1/2} \gtrsim 2n_0$, a
topology change mimicking the baryon-quark continuity takes place at $n_{1/2}$.
In terms of skyrmions, this corresponds to the transition from skyrmions to
half-skyrmions and impacts on the equation of state of dense baryonic matter.
The emergence in medium at $n_{1/2}$ of parity-doublet symmetry -- which is
invisible in QCD in matter-free vacuum -- plays the crucial role. The
consequence of the topology change is that massive compact stars carry the
"pseudo-conformal sound velocity" $v_s^2/c^2\approx 1/3$ at $n\gsim n_{1/2}$
signaling a precursor to precocious emergence of scale symmetry as well as {a}
local symmetry hidden in QCD in the matter-free vacuum. A highly significant
prediction of this work is that the topology change density from normal matter
to half-skyrmion matter, up-to-date inaccessible either by QCD proper or by
terrestrial experiments, could possibly be pinned down within the range $2 <
n_{1/2}/n_0 < 4$, commensurate with the range expected for a continuous
hadrons-to-quarks/gluons transition.
