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

Using 10.1 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 $X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1,2}$, where the $X(3872)$ is produced in $e^{+}{e}^{-}\to \gamma X(3872)$. No evidence above $3\sigma$ is found for either decay. Upper limits at the $90\mathrm{\%}$ C.L. on the branching fractions of $X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1,2}$ normalized to the branching fraction of $X(3872)\to {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi$ are set to be $\mathcal{B}(X(3872)\to {\pi }^0{\pi }^0{\chi }_{c1})\mathrm{/}\mathcal{B}(X(3872)\to {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi )<1.1$ and $\mathcal{B}(X(3872)\to {\pi }^0{\pi }^0{\chi }_{c2})\mathrm{/}\mathcal{B}(X(3872)\to {\pi }^{+}{\pi }^{-}J\mathrm{/}\psi )<0.5$, taking into account both statistical and systematic uncertainties.