  This paper completes the analysis initiated in the companion work
arXiv:2403.02283 -- referred to as Paper I -- by showing how Vasiliev's 4D
higher-spin gravity (HSG) and 3D coloured conformal matter fields coupled to
conformal higher-spin gauge fields and colour gauge fields (coloured conformal
HSG, or CCHSG) emerge as consistent reductions of a common parent model. The
latter is a Frobenius-Chern-Simons model with superconnection valued in a
fractional-spin extension of Vasiliev's higher-spin algebra, and was defined
and studied in Paper I. Here, we i) realize HSG as a subcase of a more general
4D reduction, describing HSG coupled to coloured, fractional-spin matter, which
we refer to as 4D fractional-spin gravity; ii) study the CCHSG model, in
particular exhibiting the crucial role played by novel colour gauge fields in
coupling conformal matter to conformal HSG, thereby completing the models due
to Vasiliev and Nilsson; iii) extract conformal currents and composite coloured
sources in an expansion of the CCHSG model around 3D Minkowskian leaves; and
iv) compare our results with Vasiliev's holography proposal of arXiv:1203.5554.
The common origin of HSG and CCHSG, besides relating the two models directly,
is the starting point for establishing the holographic correspondence between
the two models via overlap conditions, to be presented separately.
