  We propose a hypothesis on the detailed structure for the representation of
the conformal symmetry breaking term in the basic Crewther relation generalized
in the perturbation theory framework in QCD renormalized in the ${\rm
\bar{MS}}$ scheme. We establish the validity of this representation in the
$O(\alpha_s^4)$ approximation. Using the variant of the generalized Crewther
relation formulated here allows finding relations between specific
contributions to the QCD perturbation series coefficients for the flavor
nonsinglet part of the Adler function $D^{ns}_A$ for the electron-positron
annihilation in hadrons and to the perturbation series coefficients for the
Bjorken sum rule $S_\text{Bjp}$ for the polarized deep-inelastic lepton-nucleon
scattering. We find new relations between the $\alpha_s^4$ coefficients of
$D^{ns}_A$ and $S_\text{Bjp}$. Satisfaction of one of them serves as an
additional theoretical verification of the recent computer analytic
calculations of the terms of order $\alpha_s^4$ in the expressions for these
two quantities.
