  Current cosmological constraints on the scalar spectral index of primordial
fluctuations $n_{\rm s}$ in the $\Lambda$CDM model have excluded the minimal
scale-invariant Harrison-Zel'dovich model ($n_{\rm s}=1$; hereafter HZ) at high
significance, providing support for inflation. In recent years, however, some
tensions have emerged between different cosmological datasets that, if not due
to systematics, could indicate the presence of new physics beyond the
$\Lambda$CDM model. In the light of these developments, we evaluate the
Bayesian evidence against HZ in different data combinations and model
extensions. Considering only the Planck temperature data, we find inconclusive
evidence against HZ when including variations in the neutrino number $N_{\rm
eff}$ and/or the Helium abundance $Y_{\rm He}$. Adding the Planck polarization
data, on the other hand, yields strong evidence against HZ in the extensions we
considered. Perhaps most interestingly, Planck temperature data combined with
local measurements of the Hubble constant give as the most probable model an HZ
spectrum, with additional neutrinos. However, with the inclusion of
polarisation, standard $\Lambda$CDM is once again preferred, but the HZ model
with extra neutrinos is not strongly disfavored. The possibility of fully
ruling out the HZ spectrum is therefore ultimately connected with the solution
to current tensions between cosmological datasets. If these tensions are
confirmed by future data, then new physical mechanisms could be at work and an
HZ spectrum could still offer a valid alternative.
