  In prerecombination resolutions of the Hubble tension, such as early dark
energy, new physics before recombination shifts the values of relevant
cosmological parameters so that the models can fit with cosmic microwave
background and baryon acoustic oscillations observations as well as
$\Lambda$CDM does. In this paper, we clarify how the parameter shifts are
related with $\delta H_0$, particularly we find the shift of primordial scalar
spectral index scales as ${\delta n_s}\simeq 0.4{\delta H_0\over H_0}$ by
performing the Monte Carlo Markov chain analysis with the
Planck2018+BAO+Pantheon+R19+Keck Array/BICEP dataset. A novel point of our
result is that if the current $H_0$ measured locally is correct, complete
resolution of the Hubble tension seems to be pointing to a scale invariant
Harrison-Zeldovich spectrum, i.e. $n_s= 1$ for $H_0\sim 73$km/s/Mpc.
