  Recently, there have been increasingly hot debates on whether there exists a
quantum spin liquid in the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ in high
magnetic field. To investigate this issue, we perform the ultralow-temperature
thermal conductivity measurements on the single crystals of $\alpha$-RuCl$_3$
down to 80 mK and up to 9 T. Our experiments clearly show a field-induced phase
transition occurring at $H_c$ $\approx$ 7.5 T, above which the zigzag magnetic
order is completely suppressed. The minimum of thermal conductivity at 7.5 T is
attributed to the strong scattering of phonons by the magnetic fluctuations.
Most importantly, above 7.5 T, we do not observe any significant contribution
of thermal conductivity from gapless magnetic excitations, which puts a strong
constraint on the nature of the high-field phase of $\alpha$-RuCl$_3$.
