In a recent experiment [RB810044] we have measured the mean kinetic energy in supercooled bulk water at three pressures values in the range 200-4000 bar in order to shed light on the origin of the anomalous energy excess of water protons observed at ambient pressure around 270 K. Data obtained at T=268 K and pressures of P = 260, 530 and 4000 bar (i.e. changing the density from 0.0338 to 0.0381 molecules/A^3) seem to show the same pressure dependence of the self-diffusion coefficient of supercooled water at 268 K. Importantly, this macroscopic dynamical property shows a maximum at P=1500 bar, but the limited number of available DINS data do not allow to conclude on the possible connection between the proton environment, and in particular of the proton mean kinetic energy, and the water self diffusion coefficient. We request beam time on Vesuvio to tackle this issue.