Quantum-correlated measurements of the decay $D^{0}~\to~K^{0}_{\rm S}\pi^{+}\pi^{-⁠}\pi^{0}$ are performed with 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 $B^{\pm}~\to~D(K^{0}_{\rm S}\pi^{+}\pi^{-}\pi^{0})K^{\pm}$ decays using data collected by the Belle detector is also shown.