  We present first results for the $2 \to 4$ single-parton scattering $g g \to
c \bar c c \bar c$ subprocess for the first time fully within the
$k_t$-factorization approach. In this calculation we have used the
Kimber-Martin-Ryskin unintegrated gluon distribution which effectively includes
some class of higher-order gluon emissions, and an off-shell matrix element
squared calculated using recently developed techniques. The results are
compared with our earlier result obtained within the collinear-factorization
approach. Only slightly larger cross sections are obtained than in the case of
the collinear approach. Inclusion of transverse momenta of gluons entering the
hard process leads to a much stronger azimuthal decorrelation between $c c$ and
$\bar c \bar c$ than in the collinear-factorization approach. A comparison to
predictions of double parton scattering (DPS) results and the LHCb data
strongly suggests that the assumption of two fully independent DPS ($g g \to c
\bar c \otimes g g \to c \bar c$) may be too approximate.
