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We investigate a high-$z$ cosmological model to compute the co-moving soundhorizon $r_s$ at baryon-velocity freeze-out towards the end of hydrogenrecombination. This model assumes a replacement of the conventional CMB photongas by deconfining SU(2) Yang-Mills thermodynamics, three flavours of masslessneutrinos ($N_\nu=3$), and a {\sl purely baryonic} matter sector (no colddark-matter (CDM)). The according SU(2) temperature-redshift relation of theCMB is contrasted with recent measurements appealing to the thermalSunyaev-Zel'dovich effect and CMB-photon absorption by molecular rotationsbands or atomic hyperfine levels. Relying on a realistic simulation of theionization history throughout recombination, we obtain $z_*=1693.55\pm 6.98$and $z_{\rm drag}=1812.66\pm 7.01$. Due to considerable widths of thevisibility functions in the solutions to the associated Boltzmann hierarchy andEuler equation we conclude that $z_*$ and $z_{\rm drag}$ over-estimate theredshifts for the respective photon and baryon-velocity freeze-out. Realisticdecoupling values turn out to be $z_{{\rm lf},*}=1554.89\pm 5.18$ and $z_{\rmlf,drag}=1659.30\pm 5.48$. With $r_s(z_{\rm lf,drag})=(137.19\pm 0.45)\,$Mpcand the essentially model independent extraction of $r_s\cdot H_0=\mbox{const}$from low-$z$ data in arXiv:1607.05617 we obtain a good match with the value$H_0=(73.24\pm 1.74)\,$km\,s$^{-1}$\,Mpc$^{-1}$ extracted in arXiv:1604.01424by appealing to Cepheid calibrated SNe~Ia, new parallax measurements, strongerconstraints on the Hubble flow, and a refined computation of distance toNGC4258 from maser data. We briefly comment on a possible interpolation of ourhigh-$z$ model, invoking percolated and unpercolated U(1) topological solitonsof a Planck-scale axion field, to the phenomenologically successful low-$z$$\Lambda$CDM cosmology.

SU(2)$_{\tiny\mbox{CMB}}$ at high redshifts and the value of $H_0$