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It has been observed that conventional renormalization scheme and scaleambiguities for the pQCD predictions can be eliminated by using the principleof maximum conformality (PMC). However, being the intrinsic nature of anyperturbative theory, there are still two types of residual scale dependencesdue to uncalculated higher-order terms. In the paper, as a step forward of ourprevious work [Phys.Rev.D {\bf 89},116001(2014)], we reanalyze the electroweak$\rho$ parameter by using the PMC single-scale approach. Using the PMCconformal series and the Pad$\acute{e}$ approximation approach, we observe thatthe residual scale dependence can be greatly suppressed and then a more precisepQCD 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 aresquared averages of those from unknown higher-order terms and$\Delta\alpha_s(M_Z)=\pm 0.0010$. We then predict the magnitudes of the shiftsof the $W$-boson mass and the effective leptonic weak-mixing angle: $\deltaM_{W}|_{\rm N^4LO} =-0.26$ MeV and $\delta \sin^2{\theta}_{\rm eff}|_{\rmN^4LO}=0.14\times10^{-5}$, which are well below the precision anticipated forthe future electron-position colliders such as FCC, CEPC and ILC. Thus bymeasuring those parameters, it is possible to test SM with high precision.

A new analysis of the pQCD contributions to the electroweak parameter $\rho$ using the single-scale approach of principle of maximum conformality