  Considering the results of recent distinguished analytical calculations of
the 5-loop single-fermion loop corrections to the QED $\beta$-function we
emphasize that to our point of view it is important to perform their
independent cross-checks. We propose one of the ways of these cross-check. It
is based on the application of the original Crewther relation. We derive the
new analytical expressions for the $C_F^4\alpha_s^4$-contributions to the
Bjorken polarized sum rule. If results of possible direct calculations will
agree with the presented expression, then the appearance of $\zeta_3$-term in
the 5-loop correction to the QED $\beta$-function and in the $C_F^4\alpha_s^4$
contribution into the $e^+e^-$ annihilation Adler function will get independent
support and may be analysed within the framework of the recently introduced
concept of "maximal transcendentality".
