  The Principle of Maximum Conformality (PMC) eliminates QCD renormalization
scale-setting uncertainties using fundamental renormalization group methods.
The resulting scale-fixed pQCD predictions are independent of the choice of
renormalization scheme and show rapid convergence. The coefficients of the
scale-fixed couplings are identical to the corresponding conformal series with
zero $\beta$-function. Two all-orders methods for systematically implementing
the PMC-scale setting procedure for existing high order calculations are
discussed in this article. One implementation is based on the PMC-BLM
correspondence \mbox{(PMC-I)}; the other, more recent, method \mbox{(PMC-II)}
uses the ${\cal R}_\delta$-scheme, a systematic generalization of the minimal
subtraction renormalization scheme. Both approaches satisfy all of the
principles of the renormalization group and lead to scale-fixed and
scheme-independent predictions at each finite order. In this work, we show that
PMC-I and PMC-II scale-setting methods are in practice equivalent to each
other. We illustrate this equivalence for the four-loop calculations of the
annihilation ratio $R_{e^+ e^-}$ and the Higgs partial width $\Gamma(H\to
b\bar{b})$. Both methods lead to the same resummed (`conformal') series up to
all orders. The small scale differences between the two approaches are reduced
as additional renormalization group $\{\beta_i\}$-terms in the pQCD expansion
are taken into account. We also show that {\it special degeneracy relations},
which underly the equivalence of the two PMC approaches and the resulting
conformal features of the pQCD series, are in fact general properties of
non-Abelian gauge theory.
