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The chiral triplet pairing scenario proposed for Sr₂RuO₄ has been challenged by recent nuclear magnetic resonance (NMR) studies [A. Pustogow et al., arXiv:1904.00047 and K. Ishida et al., arXiv:1907.12236]. We perform a detailed theoretical study of spin-fluctuation mediated superconductivity guided by the spin-fluctuation spectrum measured from neutron scattering of this compound. Nodal even-parity solutions as well as odd-parity states with spins aligned predominantly out of the RuO₂ planes are found, both of which are compatible with the new data. The usual odd-parity state with spins primarily in the plane, the chiral k_x+ik_y, is difficult to stabilize and in contradiction to both NMR and neutron experiments. The presence of nodes in the spectral gap appears as a common feature for both even- and odd parity gaps. A surprising near-degeneracy of the nodal s′ and d_x²-y²-wave solutions suggests the possibility of a near-nodal time-reversal symmetry broken s′ +id_x²-y² pair state. Finally we discuss local signatures of such a state near nonmagnetic disorder, as well as the possibility of induced SDW in vicinity of impurities and under strain.