Do joint CMB and HST data support a scale invariant spectrum?

We combine current measurements of the local expansion rate, $H_0$, and Big Bang Nucleosynthesis (BBN) estimates of helium abundance with the latest cosmic microwave background (CMB) data from the Planck Collaboration to discuss the observational viability of the scale invariant Harrison-Zeldovch-Peebles (HZP) spectrum. We also analyze some of its extensions, namely, HZP + $Y_P$ and HZP + $N_{eff}$, where $Y_P$ is the primordial helium mass fraction and $N_{eff}$ is the effective number of relativistic degrees of freedom. We perform a Bayesian analysis and show that the latter model is favored with respect to the standard cosmology for values of $N_{eff}$ lying in the interval $3.70 \pm 0.13$ ($1\sigma$), which is currently allowed by some independent analyses.