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We study a generalization of the Holst action where we admit nonmetricity andtorsion in manifolds with timelike boundaries (both in the metric and tetradformalism). We prove that its space of solutions is equal to the one of thePalatini action. Therefore, we conclude that the metric sector is in factidentical to GR, which is defined by the Einstein-Hilbert action. We furtherprove that, despite defining the same space of solutions, the Palatini and (thegeneralized) Holst Lagrangians are not cohomologically equal. Thus, thepresymplectic structure and charges provided by the Covariant Phase Spacemethod might differ. However, using the relative bicomplex framework, we showthe covariant phase spaces of both theories are equivalent (and in factequivalent to GR), as well as their charges, clarifying some open problemsregarding dual charges and their equivalence in different formulations.

On the on-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries