  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.
