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