According to the 2000 claim of Lebowitz et al: ”There is however at present no rigorous mathematical derivation of Fourier’s law of heat transfer for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution”. The 2016 piston model of Bálint-Nándori-Sz.-Tóth aimed at resolving the issue. In this model, once the completion of the dynamical part of Gaspard-Gilbert’s 2008 program provides a Markov jump process for the energies of the particles in the rare interaction limit (cf. B-N-Sz-T, 2026), its hydrodynamic limit is expected to provide the heat equation. For Gaspard-Gilbert’s original 2008 model the necessary lower bound for the spectral gap for proving the hydrodynamic limit has been settled by Sasada (2015) and Carlen-Posta-Tóth (2025). For the piston model the lower bound for the spectral gap is an intriguing open problem. Likewise, is the hydrodynamic limit itself.