  Collective effects, such as Dicke superradiant emission, can enhance the
performance of a quantum device. Here, we study the heat current flowing
between a cold and a hot bath through an ensemble of $N$ qubits, which are
collectively coupled to the thermal baths. We find a regime where the
collective coupling leads to a quadratic scaling of the heat current with $N$
in a finite-size scenario. Conversely, when approaching the thermodynamic
limit, we prove that the collective scenario exhibits a parametric enhancement
over the non-collective case. We then consider the presence of a third
uncontrolled {\it parasitic} bath, interacting locally with each qubit, that
models unavoidable couplings to the external environment. Despite having a
non-perturbative effect on the steady-state currents, we show that the
collective enhancement is robust to such an addition. Finally, we discuss the
feasibility of realizing such a Dicke heat valve with superconducting circuits.
Our findings indicate that in a minimal realistic experimental setting with two
superconducting qubits, the collective advantage offers an enhancement of
approximately $10\%$ compared to the non-collective scenario.
