  We study a generalization of the Holst action where we admit nonmetricity and
torsion in manifolds with timelike boundaries (both in the metric and tetrad
formalism). We prove that its space of solutions is equal to the one of the
Palatini action. Therefore, we conclude that the metric sector is in fact
identical to GR, which is defined by the Einstein-Hilbert action. We further
prove that, despite defining the same space of solutions, the Palatini and (the
generalized) Holst Lagrangians are not cohomologically equal. Thus, the
presymplectic structure and charges provided by the Covariant Phase Space
method might differ. However, using the relative bicomplex framework, we show
the covariant phase spaces of both theories are equivalent (and in fact
equivalent to GR), as well as their charges, clarifying some open problems
regarding dual charges and their equivalence in different formulations.
