Recently, a novel material with bilayer kagome lattice Ca$_{10}$Cr$_7$O$_{28}$ was proposed to be a gapless quantum spin liquid, due to the lack of long-range magnetic order and the observation of broad diffuse excitations. Here, we present the ultralow-temperature thermal conductivity measurements on single crystals of Ca$_{10}$Cr$_7$O$_{28}$ to detect its low-lying magnetic excitations. At finite temperatures, with increasing the magnetic fields, the thermal conductivity exhibits a clear dip around 6 T, which may correspond to a crossover in the magnetic ground state. At the zero-temperature limit, no residual linear term is found at any fields, indicating the absence of gapless itinerant fermionic excitations. Therefore, if the spinons do exist, they are either localized or gapped. In the gapped case, the fitting of our data gives a small gap $Delta sim$ 0.27(2) K. These results put strong constraints on the theoretical description of the ground state in this quantum spin liquid candidate.
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