  Quantum-correlated measurements of the decay $D^{0} \to K^{0}_{\rm
S}\pi^{+}\pi^{-}\pi^{0}$ are performed using a data sample corresponding to an
integrated luminosity of 818 pb$^{-1}$ collected at the $\psi(3770)$ resonance
by the CLEO-c detector. Preliminary results are presented for the $CP$-even
fraction $F_{+}$ and the strong-phase differences of this decay. The value of
$F_{+}$ is measured to be 0.246 $\pm$ 0.018. The strong-phase differences are
measured in different regions of $K^{0}_{\rm S}\pi^{+}\pi^{-}\pi^{0}$ phase
space by binning around the intermediate resonances present. The potential
sensitivity of the results for determining the CKM angle $\gamma$ from
$B^{\pm}~\to~D(K^{0}_{\mathrm S}\pi^{+}\pi^{-}\pi^{0})K^{\pm}$ decay using the
data collected by the Belle detector is also shown.
