  The dense compact star matter is studied by using the skyrmion crystal
approach. The chiral effective theory used includes the lightest scalar meson,
the lowest-lying vector mesons as well as pions. Consistency with the vector
manifestation and the dilaton limit fixed point at high density constrains the
anomalous dimension of the gluon field $1.0 \lesssim |\gamma_{G^2}| \lesssim
2.0$ and leads to the significance of the scale symmetry breaking in the
intrinsic parity-odd part of the effective theory. The speed of sound $v_s^2
\simeq 1/3$ and the polytropic index $\gamma \simeq 1$ -- both satisfy the
conformal limits -- after the dilaton limit fixed point at high density but the
matter is still in the hadronic phase. This means that neither the conformal
speed of sound nor the smallness of the polytropic index can be used as a
criterion of the onset of quark matter. These conclusions are significant for
constructing the equation of state of nuclear matter.
