Context. The cosmological concordance model (ΛCDM) is the current standard model in cosmology thanks to its ability to reproduce the observations. The first observational evidence for this model appeared roughly 20 years ago from the type-Ia supernovae (SNIa) Hubble diagram from two different groups. However, there has been some debate in the literature concerning the statistical treatment of SNIa, and their stature as proof of cosmic acceleration.Aims. In this paper we relax the standard assumption that SNIa intrinsic luminosity is independent of redshift, and examine whether it may have an impact on our cosmological knowledge and more precisely on the accelerated nature of the expansion of the universe.Methods. To maximise the scope of this study, we do not specify a given cosmological model, but we reconstruct the expansion rate of the universe through a cubic spline interpolation fitting the observations of the different cosmological probes: SNIa, baryon acoustic oscillations (BAO), and the high-redshift information from the cosmic microwave background (CMB).Results. We show that when SNIa intrinsic luminosity is not allowed to vary as a function of redshift, cosmic acceleration is definitely proven in a model-independent approach. However, allowing for redshift dependence, a nonaccelerated reconstruction of the expansion rate is able to fit, at the same level of ΛCDM, the combination of SNIa and BAO data, both treating the BAO standard ruler rd as a free parameter (not entering on the physics governing the BAO), and adding the recently published prior from CMB observations. We further extend the analysis by including the CMB data. In this case we also consider a third way to combine the different probes by explicitly computing rd from the physics of the early universe, and we show that a nonaccelerated reconstruction is able to nicely fit this combination of low- and high-redshift data. We also check that this reconstruction is compatible with the latest measurements of the growth rate of matter perturbations. We finally show that the value of the Hubble constant (H0) predicted by this reconstruction is in tension with model-independent measurements.Conclusions. We present a model-independent reconstruction of a nonaccelerated expansion rate of the universe that is able to fit all the main background cosmological probes nicely. However, the predicted value of H0 is in tension with recent direct measurements. Our analysis points out that a final reliable and consensual value for H0 is critical to definitively prove cosmic acceleration in a model-independent way.Key words: cosmology: observations / cosmological parameters / supernovae: individual: SNIa luminosity evolution