  It has been observed that conventional renormalization scheme and scale
ambiguities for the pQCD predictions can be eliminated by using the principle
of maximum conformality (PMC). However, being the intrinsic nature of any
perturbative theory, there are still two types of residual scale dependences
due to uncalculated higher-order terms. In the paper, as a step forward of our
previous work [Phys.Rev.D {\bf 89},116001(2014)], we reanalyze the electroweak
$\rho$ parameter by using the PMC single-scale approach. Using the PMC
conformal series and the Pad$\acute{e}$ approximation approach, we observe that
the residual scale dependence can be greatly suppressed and then a more precise
pQCD prediction up to ${\rm N^4LO}$-level can be achieved, e.g.
$\Delta\rho|_{\rm PMC}\simeq(8.204\pm0.012)\times10^{-3}$, where the errors are
squared averages of those from unknown higher-order terms and
$\Delta\alpha_s(M_Z)=\pm 0.0010$. We then predict the magnitudes of the shifts
of the $W$-boson mass and the effective leptonic weak-mixing angle: $\delta
M_{W}|_{\rm N^4LO} =-0.26$ MeV and $\delta \sin^2{\theta}_{\rm eff}|_{\rm
N^4LO}=0.14\times10^{-5}$, which are well below the precision anticipated for
the future electron-position colliders such as FCC, CEPC and ILC. Thus by
measuring those parameters, it is possible to test SM with high precision.
