Leading superconducting instabilities in three-dimensional models for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>RuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math>

The unconventional superconductor ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ has been the subject of enormous interest over more than two decades, but until now form its order parameter not explicitly determined. Since groundbreaking NMR experiments revealed recently that pairs are dominant spin-singlet character, attention focused on time-reversal symmetry breaking linear combinations $s$-, $d$-, and $g$-wave one-dimensional (1D) irreducible representations. However, a state ${d}_{xz}+i{d}_{yz}$ corresponding to two-dimensional representation ${E}_{g}$ also proposed based some experiments. We present systematic study stability various superconducting candidate states, assuming pairing is driven by fluctuation exchange mechanism, including realistic three-dimensional Fermi surface, full treatment both local nonlocal spin-orbit couplings, wide range Hubbard-Kanamori interaction parameters $U,J,{U}^{\ensuremath{'}},{J}^{\ensuremath{'}}$. leading instabilities found exhibit nodal even-parity ${A}_{1g}({s}^{\ensuremath{'}})$ or ${B}_{1g}({d}_{{x}^{2}\ensuremath{-}{y}^{2}})$ symmetries, similar findings in models without longer-range Coulomb which tends favor ${d}_{xy}$ ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. Within so-called Hund's coupling mean-field scenario, ${E}_{g}({d}_{xz}/{d}_{yz})$ solution can be stabilized for large $J$ specific forms coupling, all cases studied here eigenvalues other solutions significantly larger when vertex included kernel. Additionally, we compute spin susceptibility relevant phases compare recent neutron scattering nuclear magnetic resonance (NMR) Knight shift measurements. It supports phase, contrast experiment [K. Jenni et al., Phys. Rev. B 103, 104511 (2021)], whereas ${s}^{\ensuremath{'}}+i{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ does not. Furthermore, comparison reveals exhibits low-temperature ${d}_{xz}+i{d}_{yz}$.