Search for $X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c1,2}$

Using $10.1\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data collected by the BESIII detector with center-of-mass energies between 4.15 GeV and 4.30 GeV, we search for the decays $X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c1,2}$, where the $X\left(3872\right)$ is produced in ${e}^{+}{e}^{-}\to \gamma X\left(3872\right)$. No evidence above $3\sigma $ is found for either decay. Upper limits at the 90% CL on the branching fractions of $X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c1,2}$ normalized to the branching fraction of $X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi $ are set to be $B\left(X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c1}\right)/B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)<1.1$ and $B\left(X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c2}\right)/B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)<0.5$, taking into account both statistical and systematic uncertainties.