  We find that, when the dilaton is implemented as a (pseudo-)Nambu-Goldstone
boson using a conformal compensator or "conformon" in a hidden gauge symmetric
Lagrangian written to $O(p^4)$ from which baryons arise as solitons, namely,
skyrmions, the vector manifestation and chiral symmetry restoration at high
density predicted in hidden local symmetry theory --- which is consistent with
Brown-Rho scaling --- are lost or sent to infinite density. It is shown that
they can be restored if in medium the behavior of the $\omega$ field is taken
to deviate from that of the $\rho$ meson in such a way that the flavor $U(2)$
symmetry is strongly broken at increasing density. The hitherto unexposed
crucial role of the $\omega$ meson in the structure of elementary baryon and
multibaryon systems is uncovered in this work. In the state of half-skyrmions
to which the skyrmions transform at a density $n_{1/2}^{} \gtrsim n_0^{}$
(where $n_0^{}$ is the normal nuclear matter density), characterized by the
vanishing (space averaged) quark condensate but nonzero pion decay constant,
the nucleon mass remains more or less constant at a value $\gtrsim$ 60 \% of
the vacuum value indicating a large component of the nucleon mass that is not
associated with the spontaneous breaking of chiral symmetry. We discuss its
connection to the chiral-invariant mass $m_0^{}$ that figures in the
parity-doublet baryon model.
