The recent report of a half-quantized thermal Hall effect in the Kitaev
material α-RuCl3 has sparked a strong debate on whether it is
generated by Majorana fermion edge currents or whether other more conventional
mechanisms involving magnons or phonons are at its origin. A more direct
evidence for Majorana fermions which could be expected to arise from a
contribution to the longitudinal heat conductivity κxx at
T→0 is elusive due to a very complex magnetic field dependence of
κxx. Here, we report very low temperature (below 1~K) thermal
conductivity (κ) of another candidate Kitaev material,
Na2Co2TeO6. The application of a magnetic field along different
principal axes of the crystal reveals a strong directional-dependent
magnetic-field (B) impact on κ. We show that no evidence for
mobile quasiparticles except phonons can be concluded at any field from 0~T to
the field polarized state. In particular, severely scattered phonon transport
is observed across the B−T phase diagram, which is attributed to prominent
magnetic fluctuations. Cascades of phase transitions are uncovered for all B directions by probing the strength of magnetic fluctuations via a precise
record of κ(B). Our results thus rule out recent proposals for
itinerant magnetic excitations in Na2Co2TeO6, and emphasise the
importance of discriminating true spin liquid transport properties from
scattered phonons in candidate materials