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We calculate the primordial power spectrum of tensor perturbations, within the emergent universe scenario, incorporating a version of the Continuous Spontaneous Localization (CSL) model as a mechanism capable of: breaking the initial symmetries of the system, generating the perturbations, and also achieving the quantum-to-classical transition of such perturbations. We analyze how the CSL model modifies the characteristics of the B-mode CMB polarization power spectrum, and we explore their differences with current predictions from the standard concordance cosmological model. We have found that, regardless of the CSL mechanism, a confirmed detection of primordial B-modes that fits to a high degree of precision the shape of the spectrum predicted from the concordance $Λ$CDM model, would rule out one of the distinguishing features of the emergent universe. Namely, achieving a best fit to the data consistent with the suppression observed in the low multipoles of the angular power spectrum of the temperature anisotropy of the CMB. On the contrary, a confirmed detection that accurately exhibits a suppression of the low multipoles in the B-modes, would be a new feature that could be considered as a favorable evidence for the emergent scenario. In addition, we have been able to establish an upper bound on the collapse parameter of the specific CSL model used.