  The De Donder-Weyl (DW) Hamiltonian theory of fields treats space and time
variables on equal footing. Its quantization, called precanonical quantization,
leads to a hypercomplex generalization of quantum formalism to field theory as
it follows from the quantization of Poisson-Gerstenhaber brackets defined on
differential forms. Our recent work on precanonical quantization of general
relativity is extended to the teleparallel equivalent of general relativity
(TEGR) in tetrad Palatini formulation. The covariant precanonical Schr\"odinger
equation for quantum TEGR and the relevant operators are constructed from the
quantization of generalized Dirac brackets calculated using the constraints
analysis generalized to the DW Hamiltonian theory. Our analysis of the ordering
ambiguities in the precanonical Schr\"odinger equation allows us to estimate
the contribution to the cosmological constant from the quantum TEGR and argue
its consistency with the observed value, albeit with the current error of
estimation of 13 orders of magnitude due to the theoretical uncertainties in
the relation between the scale $\varkappa$ introduced by precanonical
quantization and the mass gap in the pure gauge sector of QCD.
