The $Lambda$CDM model is the current standard model in cosmology thanks to its ability to reproduce the observations. Its first observational evidence appeared from the type Ia supernovae (SNIa) Hubble diagram. However, there has been some debate in the literature concerning the statistical treatment of SNIa. In this paper we relax the standard assumption that SNIa intrinsic luminosity is independent of the redshift, and we examine whether it may have an impact on the accelerated nature of the expansion of the Universe. In order to be as general as possible, we reconstruct the expansion rate of the Universe through a cubic spline interpolation fitting observations of different probes: SNIa, baryon acoustic oscillations (BAO), and the high-redshift information from the cosmic microwave background (CMB). We show that when SNIa intrinsic luminosity is not allowed to vary as a function of the redshift, cosmic acceleration is definitely proven in a model-independent approach. However, allowing for a redshift dependence, a non-accelerated reconstruction of the expansion rate is able to fit, as well as $Lambda$CDM, the combination of SNIa and BAO data, both treating the BAO standard ruler $r_d$ as a free parameter, or adding the recently published prior from CMB observations. We further extend the analysis by including the CMB data, and we show that a non-accelerated reconstruction is able to nicely fit this combination of low and high-redshift data. In this work we present a model-independent reconstruction of a non-accelerated expansion rate of the Universe that is able to nicely fit all the main background cosmological probes. However, the predicted value of $H_0$ is in tension with recent direct measurements. Our analysis points out that a final, reliable, and consensual value for $H_0$ would be critical to definitively prove the cosmic acceleration in a model-independent way. [Abridged]
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