  We calculate the finite temperature thermal conductivity of a time-reversal
invariant chiral $\mathbb{Z}_3$ clock model along an integrable line in the
parameter space using tDMRG. The thermal current itself is not a conserved
charge, unlike in the XXZ model, but has a finite overlap with a local
conserved charge $Q^{(2)}$ obtained from the transfer matrix. We find that the
Drude weight is finite at non-zero temperature, and the Mazur bound from
$Q^{(2)}$ saturates the Drude weight, allowing us to obtain an asymptotic
expression for the Drude weight at high temperatures. The numerical estimates
are validated using a sum rule for thermal conductivity. On the computational
side, we also explore the effectiveness of the ancilla disentangler in the
integrable and non-integrable regimes of the model. We find that the
disentangler helps in localizing the entanglement growth around the quench
location, but the improvement is lesser in the non-integrable regime and at low
temperatures.
