The material α-RuCl₃ has been extensively researched as a potential candidate for Kitaev quantum spin liquid, whose low-energy excitations are expected to be Majorana fermions. Note that α-RuCl₃ orders antiferromagnetically at low temperature, under T_N = 7 K, but an external magnetic field in the plane, greater than H_∥ = 7 T, removes this order. The question is whether or not there is, over a range of fields greater than 7 T, an orderless region that harbours a liquid state of quantum spin. The recent announcement of a quantized thermal Hall conductivity κ_xy at H_∥ > 7 T by Kasahara et al. [Kasahara et al., Nature 559, 227 (2018)] would be the first direct evidence of itinerant Majorana fermions in a quantum spin liquid state. In the study we present here, we examine the origin of κ_xy in α-RuCl₃. The comparison between the longitudinal thermal conductivity κ_xx and κ_xy suggests that κ_xy is mainly dominated by phonons - questioning the possibility of a quantified κ_xy. In addition, depending on H_∥, we do not see an intermediate region between the antiferromagnetic state under 7 T and the state of polarized spins at strong magnetic field, thus leaving little room for a quantum spin liquid state.