  The quintessence dark energy potential is reconstructed in a
model-independent way. Reconstruction relies on a Gaussian process and on
available expansion-rate data. Specifically, 40-point values of $H(z)$ are
used, consisting of a 30-point sample deduced from a differential age method
and an additional 10-point sample obtained from the radial BAO method. Results
are obtained for two kernel functions and for three different values of
$H_{0}$. They shed light on the $H_{0}$ tension problem for a universe
described with quintessence dark energy. They are also a clear indication that
the tension has to do with the physical understanding of the issue, rather than
being just a numerical problem with statistics. Moreover, the model-independent
reconstruction of the potential here obtained can serve as a reference to
constraint available models and it can be also used as a reference frame to
construct new ones. Various possibilities, including $V(\phi) \sim e^{-\lambda
\phi}$, are compared with the reconstructions here obtained, which is notably
the first truly model independent reconstruction of the quintessence dark
energy potential. This allows to select new models that can be interesting for
cosmology. The method can be extended to reconstruct the potential of related
dark energy models, to be considered in future work.
