We investigate proton-neutron ($pn$) pairing correlations within a generalized Hartree-Fock-Bogoliubov (HFB) framework using the Gogny D1S energy density functional. When $pn$ mixing is allowed, large single-particle bases trigger numerical instabilities in Gogny D1S due to its zero-range density-dependent term contribution to the $pn$ pairing field, whereas stable solutions are obtained with the Hamiltonian-based Brink-Boecker B1 interaction. Constrained HFB calculations in reduced spaces show that self-consistent minima with Gogny D1S systematically correspond to vanishing $pn$ pairing with total energy rising rapidly as $pn$ pairing correlations are introduced. Furthermore, particle-number restored calculations significantly flatten these HFB energy curves, inducing $pn$ mixing across the collective coordinates within the variation after particle-number projection framework. These results provide key insights into the limitations and behavior of Gogny functionals under generalized HFB conditions, offering guidance for future functional re-parametrizations and beyond-mean-field studies.

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• Nuclear forces
• Nuclear structure & decays
• Nuclear density functional theory
• Nuclear many-body theory