  We study whether in-medium showers of high-energy gluons can be treated as a
sequence of individual splitting processes $g{\to}gg$, or whether there is
significant quantum overlap between where one splitting ends and the next
begins. Accounting for the Landau-Pomeranchuk-Migdal (LPM) effect, we calculate
such overlap effects to leading order in high-energy $\alpha_{\rm s}(\mu)$ for
the simplest theoretical situation. We investigate a measure of overlap effects
that is independent of physics that can be absorbed into an effective value
$\hat q_{\rm eff}$ of the jet-quenching parameter $\hat q$.
