  We investigate thermal Hall conductivity $\kappa_{xy}$ of a $J$-$K$
Kitaev-Heisenberg model with a Zeeman field in the $(111)$ direction in the
light of the recent debate surrounding the possible re-emergence of Ising
topological order (ITO) and half-quantized $\kappa_{xy}/T$ upon
field-suppression of long-range magnetic order in Kitaev materials. We use the
purification-based finite temperature Tensor Network approach making no prior
assumptions about the nature of the excitations: Majorana, visons or spin
waves. For purely Kitaev interactions and fields $h/K \gtrsim 0.02$ sufficient
to degrade ITO, the peak $\kappa_{xy}/T$ monotonously decreases from
half-quantization associated with lower fields - a behavior reminiscent of
vison fluctuation corrections. For higher fields $h/K\gtrsim 0.1,$ we find the
results qualitatively consistent with a spin-wave treatment. In our $J$-$K$
model (with ferro-$K$ and antiferro-$J$), in the vicinity of field-suppressed
magnetic order, we found $\kappa_{xy}/T$ to be significant, with peak
magnitudes exceeding half-quantization followed by a monotonous decrease with
increasing $h.$ We thus conclude that half-quantized thermal Hall effect in the
vicinity of field suppressed magnetic order in our model, is a fine-tuning
effect and is not associated with a Majorana Hall state with ITO.
