  In this work we first discuss the possibility that dark energy models with
negative energy density values in the past can alleviate the $H_0$ tension, as
well as the discrepancy with the BAO Ly-$\alpha$ data, both which prevail
within the $\Lambda$CDM model. We then investigate whether two minimal
extensions of the $\Lambda$CDM model, together or separately, can successfully
realize such a scenario: (i) The spatial curvature, which, in the case of
spatially closed universe, mimics a negative density source. (ii)
Simple-graduated dark energy (-gDE), which promotes the null inertial mass
density of the usual vacuum energy to an arbitrary constant -- if negative, the
corresponding energy density decreases with redshift similar to the phantom
models, but unlike them crosses below zero at a certain redshift. We find that,
when the Planck data are not included in the observational analysis, the models
with simple-gDE predict interesting and some significant deviations from the
$\Lambda$CDM model. In particular, a spatially closed universe along with a
simple-gDE of positive inertial mass density, which work in contrast to each
other, results in minor improvement to the $H_0$ tension. The joint data set,
including the Planck data, presents no evidence for a deviation from spatial
flatness, but almost the same evidence for a cosmological constant and the
simple-gDE with an inertial mass density of order $\mathcal{O}(10^{-12})\,\rm
eV^4$. The latter case predicts almost no deviation from the $\Lambda$CDM model
up until today -- so that it results in no improvement regarding the BAO
Ly-$\alpha$ data -- , except that it slightly aggravates the $H_0$ tension. We
also study via dynamical analysis the history of the universe in the models, as
the simple-gDE results in futures different than the de Sitter future of the
$\Lambda$CDM model.
