High-pressure supercritical turbulence simulations are strongly susceptible to numerical instabilities. The multi-scale nature of the flow, in conjunction with the nonlinear thermodynamics and the strong density gradients across the pseudo-boiling line can trigger spurious pressure oscillations and unbounded amplification of aliasing errors. A wide variety of regularization approaches have been traditionally utilized by the community, including upwind-biased schemes, artificial dissipation, and/or high-order filtering, where stability is achieved at the expense of suppressing part of the turbulent energy spectrum. In this work, a novel numerical scheme based on the paradigm of physics-compatible discretizations is investigated. In particular, the proposed method discretely enforces kinetic-energy conservation (by convection) as well as preservation of pressure equilibrium; the former is achieved using proper splitting of the convective terms, whereas the latter is obtained by directly evolving an equation for pressure. The simultaneous enforcement of these two properties can lead to stable and physically consistent scale-resolving simulations of supercritical turbulence without the need for any form of artificial stabilization. The novel method is preliminarly assessed on two benchmark cases, with numerical results supporting the theoretical findings.