  The analytical $\mathcal{O}(a^4_s)$ perturbative QCD expression for the
flavour non-singlet contribution to the Bjorken polarized sum rule in the
rather applicable at present gauge--dependent $\rm{miniMOM}$ scheme is
obtained. For the considered three values of the gauge parameter, namely
$\xi=0$ (Landau gauge), $\xi=-1$ (anti--Feynman gauge) and $\xi=-3$
(Stefanis--Mikhailov gauge), the scheme-dependent coefficients are considerably
smaller than the gauge-independent $\rm{\overline{MS}}$ results. It is found
that the fundamental property of the factorization of the QCD renormalization
group $\beta$-function in the generalized Crewther relation, which is valid in
the gauge-invariant $\rm{\overline{MS}}$ scheme up to
$\mathcal{O}(a^4_s)$-level at least, is unexpectedly valid at the same level in
the $\rm{miniMOM}$-scheme for $\xi=0$, and for $\xi=-1$ and $\xi=-3$ in part.
