Determination of the $C\text{\hspace{0.17em}⁣}P$-even fraction of $D^0\to K_S^0{\pi }^{+}{\pi }^{-}{\pi }^0$

Quantum-correlated $D\bar{D}$ pairs collected by the BESIII experiment at the $\psi (3770)$ resonance, corresponding to an integrated luminosity of 2.93 fb${}^{-1}$, are used to study the $D^0\to K_S^0{\pi }^{+}{\pi }^{-}{\pi }^0$ decay mode. The $C\text{\hspace{0.17em}⁣}P$-even fraction of $D^0\to K_S^0{\pi }^{+}{\pi }^{-}{\pi }^0$ decays is determined to be $0.235\pm 0.010\pm 0.002$, where the first uncertainty is statistical and the second is systematic.