Recently, a new family of iron-based superconductors called 12442 was discovered and the muon spin relaxation ($mu$SR) measurements on KCa$_2$Fe$_4$As$_4$F$_2$ and CsCa$_2$Fe$_4$As$_4$F$_2$ polycrystals, two members of the family, indicated that both have a nodal superconducting gap structure with $s+d$ pairing symmetry. Here we report the ultralow-temperature thermal conductivity measurements on CsCa$_2$Fe$_4$As$_4$F$_2$ single crystals ($T_c$ = 29.3 K). A negligible residual linear term $kappa_0/T$ in zero field and the field dependence of $kappa_0/T$ suggest multiple nodeless superconducting gaps in CsCa$_2$Fe$_4$As$_4$F$_2$. This gap structure is similar to CaKFe$_4$As$_4$ and moderately doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$, but contrasts to the nodal gap structure indicated by the $mu$SR measurements on CsCa$_2$Fe$_4$As$_4$F$_2$ polycrystals.
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