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A substantial body of phenomenological and theoretical work over the last few years strengthens the possibility that the vacuum energy density (VED) of the universe is dynamical, and in particular that it adopts the ‘running vacuum model’ (RVM) form, in which the VED evolves mildly as $\delta {\rho }_{\mathrm{vac}}\left(H\right)\sim {\nu }_{\mathrm{eff}}{m}_{\mathrm{Pl}}^2\mathcal{O}\left(H^2\right)$, where H is the Hubble rate and ${\nu }_{\mathrm{eff}}$ is a (small) free parameter. This dynamical scenario is grounded on recent studies of quantum field theory (QFT) in curved spacetime and also on string theory. It turns out that what we call the ‘cosmological constant’, $\mathsf{\Lambda }$, is no longer a rigid parameter but the nearly sustained value of $8\pi G\left(H\right){\rho }_{\mathrm{vac}}\left(H\right)$ around any given epoch $H\left(t\right)$, where $G\left(H\right)$ is the gravitational coupling, which can also be very mildly running (logarithmically). Of particular interest is the possibility suggested in past works that such a running may help to cure the cosmological tensions afflicting the $\mathsf{\Lambda }$CDM. In the current study, we reanalyze the RVM in full and we find it becomes further buttressed. Using modern cosmological data, namely a compilation of the latest SNIa+BAO+$H\left(z\right)$+LSS+CMB observations, we probe to what extent the RVM provides a quality fit better than the concordance $\mathsf{\Lambda }$CDM model, with particular emphasis on its impact on the ${\sigma }_8$ and $H_0$ tensions. We utilize the Einstein–Boltzmann system solver CLASS and the Monte Carlo sampler MontePython for the statistical analysis, as well as the statistical DIC criterion to compare the running vacuum against the rigid vacuum (${\nu }_{\mathrm{eff}}=0$). On fundamental grounds, ${\nu }_{\mathrm{eff}}$ receives contributions from all the quantized matter fields in FLRW spacetime. We show that with a tiny amount of vacuum dynamics (${\nu }_{\mathrm{eff}}\ll 1$) the global fit can improve significantly with respect to the $\mathsf{\Lambda }$CDM and the mentioned tensions may subside to inconspicuous levels. Full article