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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 $Δ$, describing a possible strong-field SEP violation, identically to all other parameters. We obtain a violation parameter $Δ= (+0.5 \pm 1.8) \times 10^{-6}$ 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 $ω_{\rm BD} > 140\,000$ 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.