Determination of the $CP$-even fraction of ${D}^{0}\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}{\pi }^{0}$

Quantum-correlated $D\overline{D}$ pairs collected by the BESIII experiment at the $\psi \left(3770\right)$ resonance corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ are used to study the ${D}^{0}\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ decay mode. The $CP$-even fraction of ${D}^{0}\to {K}_{S}^{0}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ decays is determined to be $0.235±0.010±0.002$, where the first uncertainty is statistical and the second is systematic.