We study three proposals for broken symmetry in the cuprate pseudogap - oxygen antiferromagnetism, $Theta_{II}$ orbital loop currents, and circulating currents involving apex oxygens - through numerical exploration of multi-orbital Hubbard models. Our numerically exact results show no evidence for the existence of oxygen antiferromagnetic order or the $Theta_{II}$ phase in the three-orbital Hubbard model. The model also fails to sustain an ordered current pattern even with the presence of additional apex oxygen orbitals. We thereby conclude that it is difficult to stabilize the aforementioned phases in the multi-orbital Hubbard models for parameters relevant to cuprate superconductors. However, the $Theta_{II}$ phase might be stabilized through explicit flux terms. We find an enhanced propensity for circulating currents with such terms in calculations simulating applied stress or strain, which skew the copper-oxygen plane to resemble a kagome lattice. We propose an experimental viewpoint to shed additional light on this problem.
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