Not yet reviewed by Pith; the record is open.

We investigate both the longitudinal thermal conductivity ($\kappa_{xx}$) and the planar thermal Hall conductivity ($\kappa_{xy}$) in the Kitaev spin liquid candidate of Co-based honeycomb antiferromagnet Na$_2$Co$_2$TeO$_6$ in a magnetic field ($B$) applied along the $a$ and $a^*$ axes. A finite $\kappa_{xy}$ is resolved for both field directions in the antiferromagnetic (AFM) phase below the N\'eel temperature of 27 K. The temperature dependence of $\kappa_{xy}/T$ shows the emergence of topological bosonic excitations. In addition, the field dependence of $\kappa_{xy}$ shows sign reversals at the critical fields in the AFM phase, suggesting the changes in the Chern number distribution of the topological magnons. Remarkably, a finite $\kappa_{xy}$ is observed in $B \parallel a^*$ between the first-order transition field in the AFM phase and the saturation field, which is prohibited in a disordered state by the two-fold rotation symmetry around the $a^*$ axis of the honeycomb lattice, showing the presence of a magnetically ordered state that breaks the two-fold rotation symmetry. Our results demonstrate the presence of topological magnons in this compound in the whole field range below the saturation field.