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Results of an analysis of the BRITE-Constellation photometry of the SB1 system and ellipsoidal variable $π^5$ Ori (B2\,III) are presented. In addition to the orbital light-variation, which can be represented as a five-term Fourier cosine series with the frequencies $f_{\rm orb}$, $2f_{\rm orb}$, $3f_{\rm orb}$, $4f_{\rm orb}$ and $6f_{\rm orb}$, where $f_{\rm orb}$ is the system's orbital frequency, the star shows five low-amplitude but highly-significant sinusoidal variations with frequencies $f_i$ ($i ={}$2,..,5,7) in the range from 0.16 to 0.92~d$^{-1}$. With an accuracy better than 1$σ$, the latter frequencies obey the following relations: $f_2-f_4 = 2f_{\rm orb}$, $f_7 - f_3 = 2f_{\rm orb}$, $f_5 = f_3 - f_4 = f_7 - f_2$. We interpret the first two relations as evidence that two high-order $\ell = 1, m = 0$ gravity modes are self-excited in the system's tidally distorted primary component. The star is thus an ellipsoidal SPB variable. The last relations arise from the existence of the first-order differential combination term between the two modes. Fundamental parameters, derived from photometric data in the literature and the {\em Hipparcos\/} parallax, indicate that the primary component is close to the terminal stages of its main sequence (MS) evolution. Extensive Wilson-Devinney modeling leads to the conclusion that best fits of the theoretical to observed light-curves are obtained for the effective temperature and mass consistent with the primary's position in the HR diagram and suggests that the secondary is in an early MS evolutionary stage.