  An important question concerning in-medium high-energy parton showers in a
quark-gluon plasma or other QCD medium is whether consecutive splittings of the
partons in a given shower can be treated as quantum mechanically independent,
or whether the formation times for two consecutive splittings instead have
significant overlap. Various previous calculations of the effect of overlapping
formation times have either (i) restricted attention to a soft bremsstrahlung
limit, or else (ii) used the large-$N_c$ limit (where $N_c{=}3$ is the number
of quark colors). In this paper, we make a first study of the accuracy of the
large-$N_c$ limit used by those calculations of overlap effects that avoid a
soft bremsstrahlung approximation. Specifically, we calculate the $1/N_c^2$
correction to previous $N_c{=}\infty$ results for overlap $g \to gg \to ggg$ of
two consecutive gluon splittings $g \to gg$. At order $1/N_c^2$, there is
interesting and non-trivial color dynamics that must be accounted for during
the overlap of the formation times.
