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We propose a system where coherent thermal transport between two reservoirs in non-galvanic contact is modulated by independently tuning the electron-photon and the electron-phonon coupling. The scheme is based on two gate-controlled electrodes capacitively coupled through a dc-SQUID (superconducting quantum interference device) as an intermediate phase-tunable resonator. Thereby the electron-photon interaction is modulated by controlling the flux threading the dc-SQUID (superconducting quantum interference device) and the impedance of the two reservoirs, while the electron-phonon coupling is tuned by controlling the charge carrier concentration in the electrodes. To quantitatively evaluate the behavior of the system, we propose to exploit the graphene reservoirs. In this case, the scheme can work at temperatures reaching 1 K, with unprecedented temperature modulations as large as 245 mK, transmittance up to 99%, and energy conversion efficiency up to 50%. Finally, the accuracy of heat transport control allows us to use this system as an experimental tool to determine the electron-phonon coupling in two-dimensional electronic systems. [ABSTRACT FROM AUTHOR]