  We study energy transport in the integrable $\mathbb Z_3$ parafermionic chain
using the partitioning protocol. By exploiting the Bethe-ansatz solution for
the thermodynamics of the system, we develop a generalized hydrodynamic
description of the non-equilibrium steady states, which we benchmark using
numerical simulations based on matrix product states. The model features a
low-energy conformal limit with central charge $c=4/5$, which affects the
low-temperature energy current, as we explicitly show. Moreover, we exploit
that, for energies close to the maximally excited state, the system is also
critical and described by a conformal field theory with $c=1$. By considering
the two halves prepared at two temperatures both low in value but opposite in
sign, we are able to investigate in an exact and controlled way the junction
between two conformal field theories with different central charges.
Notwithstanding the absence of global conformal invariance, we find results
that approximate to a high degree those of out-of-equilibrium conformal field
theories. Our study extends the generalized hydrodynamics to a novel framework,
where it can be profitably used for exploring new physical phenomena.
