Aftershock rates typically decay with time t after the mainshock according to the Omori–Utsu law, R(t)=K(c+t)^−p⁠, with parameters K, c, and p. The rate‐and‐state (RS) model, which is currently the most popular physics‐based seismicity model, also predicts an Omori–Utsu decay with p = 1 and a c‐value that depends on the size of the coseismic stress change. Because the mainshock‐induced stresses strongly vary in space, the c‐value should vary accordingly. Short‐time aftershock incompleteness (STAI) in earthquake catalogs has prevented a detailed test of this prediction so far, but the newly developed a‐positive method for reconstructing the true earthquake rate now allows its testing. Using previously published slip models, we calculate the coseismic stress changes for the six largest mainshocks in Southern California in recent decades and estimate the maximum shear as a scalar proxy of the coseismic stress tensor. Aftershock rates reconstructed for events in different stress ranges show that the rates follow a power law with p = 1 independent of stress with no clear sign of a c‐value. The onset of the power‐law decay is abrupt and more delayed in areas with smaller stress changes. The observations do not necessarily contradict the RS model, as STAI limits the resolution for early aftershocks, and the RS model can reproduce the observations for specific Aσ values. However, the observations lead to strong constraints, namely Aσ<10  kPa and a power‐law decay of the background rate with distance to the fault, with exponent 2.7.