The spatial evolution of pickup protons moving from the termination shock outward to the heliopause, after having suffered stochastic preacceleration in the supersonic solar wind, is studied taking into account ongoing velocity diffusion of the protons in the turbulent Alfv¿nic fluctuation field. Here only the upwind part of the heliosheath is considered. It is shown that spectra of pickup protons in the inner heliosheath are substantially different from those in the region inside the termination shock. Differential fluxes of energetic neutral atoms originating from pickup protons by charge exchange with interstellar hydrogen atoms in the inner heliosheath and arriving at 1 AU are calculated. It is shown that observations of energetic neutral hydrogen fluxes in a sufficiently wide energy range can be used to place strong limits on the level of solar wind turbulence near the termination shock and further downstream.