Mechanisms of the suppression of the electron-hole exchange interaction in nonradiative excitons with a large in-plane wave vector in high-quality heterostructures with quantum wells are analyzed theoretically. It is shown that the dominant suppression mechanism is exciton-exciton scattering accompanied by the mutual spin flips of like carriers (either two electrons or two holes), comprising the excitons. As a result, the electron spin polarization in nonradiative excitons may be retained for a long time. The analysis of experimental data shows that this relaxation time can exceed one nanosecond. This long-term and optically controllable spin memory in an exciton reservoir may be of interest for future information technologies.