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Considering the results of recent distinguished analytical calculations ofthe 5-loop single-fermion loop corrections to the QED $\beta$-function weemphasize that to our point of view it is important to perform theirindependent cross-checks. We propose one of the ways of these cross-check. Itis based on the application of the original Crewther relation. We derive thenew analytical expressions for the $C_F^4\alpha_s^4$-contributions to theBjorken polarized sum rule. If results of possible direct calculations willagree with the presented expression, then the appearance of $\zeta_3$-term inthe 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 independentsupport and may be analysed within the framework of the recently introducedconcept of "maximal transcendentality".

Is it possible to check urgently the 5-loop analytical results for the $e^+e^-$-annihilation Adler function ?