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A measurement of the $CP$-even fraction of the decay ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is performed with a quantum-correlated $\ensuremath{\psi}(3770)\ensuremath{\rightarrow}D\overline{D}$ data sample collected by the BESIII experiment, corresponding to an integrated luminosity of $2.93\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Using a combination of $CP$ eigenstates, $D\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ and $D\ensuremath{\rightarrow}{K}_{S,L}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ as tagging modes, the $CP$-even fraction is measured to …