  The gravitational strong equivalence principle (SEP) is a cornerstone of the
general theory of relativity (GR). The extreme difference in binding energy
between neutron stars and white dwarfs allows for precision tests of the SEP
via the technique of pulsar timing.
  To date, the best limit on the validity of SEP under strong-field conditions
was obtained with a unique pulsar in a triple stellar system, PSR J0337+1715.
We report here on an improvement of this test using an independent data set
acquired over 6 years with the Nan\c cay radio telescope (NRT). The
improvements arise from a uniformly sampled data set, a theoretical analysis,
and a treatment that fixes some short-comings in the previously published
results, leading to better precision and reliability of the test.
  In contrast to the previously published test, we use a different long-term
timing data set, developed a new timing model and an independent numerical
integration of the motion of the system, and determined the masses and orbital
parameters with a different methodology that treats the parameter $\Delta$,
describing a possible strong-field SEP violation, identically to all other
parameters.
  We obtain a violation parameter $\Delta = (+0.5 \pm 1.8) \times 10^{-6}$ at
95\% confidence level, which is compatible with and improves upon the previous
study by 30\%. This result is statistics-limited and avoids limitation by
systematics as previously encountered. We find evidence for red noise in the
pulsar spin frequency, which is responsible for up to 10\% of the reported
uncertainty. We use the improved limit on SEP violation to place constraints on
a class of well-studied scalar-tensor theories, in particular we find
$\omega_{\rm BD} > 140\,000$ for the Brans-Dicke parameter. The conservative
limits presented here fully take into account current uncertainties in the
equation for state of neutron-star matter.
