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Herein we study the inverse problem on inferring depth profile of near-surface residual stress in a weakly anisotropic medium by boundary measurement of Rayleigh-wave dispersion if all other relevant material parameters of the elastic medium are known. Our solution of this inverse problem is based on a recently developed algorithm by which each term of a high-frequency asymptotic formula for dispersion relations can be computed for Rayleigh waves that propagate in various directions along the free surface of a vertically-inhomogeneous, prestressed, and weakly anisotropic half-space. As a prime example of possible applications we focus on a thick-plate sample of AA 7075-T651 aluminum alloy, which has one face treated by low plasticity burnishing (LPB) that induced a depth-dependent prestress at and immediately beneath the treated surface. We model the sample as a prestressed, weakly-textured orthorhombic aggregate of cubic crystallites and assume that by nondestructive and/or destructive measurements we have ascertained everything about the sample, including the LPB-induced prestress, before it is put into service. Under the supposition that the prestress be partially relaxed but other material parameters remain unchanged after the sample undergoes a period of service, we examine the possibility of inferring the depth profile of the partially relaxed stress by boundary measurement of Rayleigh-wave dispersion.