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The present knowledge of the renormalon structure of Bjorken polarised sumrule (BSR) ${\overline \Gamma}_1^{{\rm p-n}}(Q^2)$ is used to construct itscharacteristic function and thus to evaluate the leading-twist part of BSR. Inthis resummation, we take for the running coupling either the perturbative QCD(pQCD) coupling $a(Q^2) \equiv \alpha_s(Q^2)/\pi$ in specific schemes, orholomorphic couplings [$(a(Q^2) \mapsto {\mathcal A}(Q^2)$] that have no Landausingularities. The $D=2$ and $D=4$ terms are included in the Operator ProductExpansion (OPE) of inelastic BSR, and fits are performed to the availableexperimental data in various intervals $(Q^2_{\rm min},Q^2_{\rm max})$ where $Q^2_{\rm max}=4.74 \ {\rm GeV}^2$. Since the pQCD coupling $a(Q^2)$ has Landausingularities at $Q^2 \lesssim 1 \ {\rm GeV}^2$, we need relatively high$Q^2_{\rm min} \approx 1.7 \ {\rm GeV}^2$ in the pQCD case. When holomorphiccouplings ${\mathcal A}(Q^2)$ are used, no such problems occur; for the $3\delta$AQCD and $2 \delta$AQCD variants the preferred values are $Q^2_{\rm min}\approx 0.6 \ {\rm GeV}^2$. The preferred values of $\alpha_s$ in generalcannot be unambiguously extracted, due to large uncertainties of theexperimental BSR data, although the fit with $2 \delta$AQCD and the $D=2$ termincluded suggests $\alpha_s^{{\overline {\rm MS}}}(M_Z^2) \approx 0.1181$. Atfixed value of $\alpha_s^{{\overline {\rm MS}}}(M_Z^2)$, the values of the$D=2$ and $D=4$ residue parameters are determined in all cases, with thecorresponding uncertainties.

Renormalon-based resummation of Bjorken polarised sum rule in perturbative and holomorphic QCD