CFD simulation has become widespread in the design of cost-efficient polymer manufacturing processes. However, flow simulations require the right constitutive equations governing the complex rheological and thermo-physical properties of polymeric materials. Over the last decades, significant progress has been made to implement macroscopic network models (eXtended Pom-Pom, Rolie-Poly…) into CFD packages. Surprisingly, the inherent non-isothermal character of polymer processing and the strong coupling between the thermo-physical properties, like thermal conductivity (TC), and the rheological behavior is often neglected. The modelling of thermo-physical properties is limited by the poor understanding of the mechanisms of phonon transport in amorphous polymers. We present a Molecular Dynamics (MD) study of the anisotropy in TC in polyethylene and polystyrene melts subjected to deformation. Simultaneous characterization of the stress-strain behavior, a piece missing in previous simulation work, has allowed us to obtain the first MD confirmation of the validity of a linear relationship between TC and stress tensors known as the stress-thermal rule (STR). In agreement with the available experimental and simulation data, we find that TC increases (decreases) in the direction parallel (perpendicular) to the stretch. However, our results for polyethylene dispute the STR universality previously suggested by experiments (i.e., independence of polymer chemistry). Finally, we test the key hypothesis of preferential heat transport along the polymer chain backbone by establishing the TC dependence on chain length. We observe that TC increases with chain length until the system becomes entangled. Overall, our results highlight the importance of entanglement interactions to explain thermal transport in polymer melts. Our findings present a stepping stone to develop a robust molecular-to-continuum methodology to study the non-isothermal flows present in polymer manufacturing.