  Based on a formalism introduced in our previous work, we reconstruct the
phenomenological function $G_{\rm eff}(z)$ describing deviations from General
Relativity (GR) in a model-independent manner. In this alternative approach, we
model $\mu\equiv G_\mathrm{eff}/G$ as a Gaussian process and use forecasted
growth-rate measurements from a stage-IV survey to reconstruct its shape for
two different toy models. We follow a two-step procedure: (i) we first
reconstruct the background expansion history from Supernovae (SNe) and Baryon
Acoustic Oscillation (BAO) measurements; (ii) we then use it to obtain the
growth history $f\sigma_8$, that we fit to redshift-space distortions (RSD)
measurements to reconstruct $G_\mathrm{eff}$. We find that upcoming surveys
such as the Dark Energy Spectroscopic Instrument (DESI) might be capable of
detecting deviations from GR, provided the dark energy behavior is accurately
determined. We might even be able to constrain the transition redshift from
$G\to G_\mathrm{eff}$ for some particular models. We further assess the impact
of massive neutrinos on the reconstructions of $G_\mathrm{eff}$ (or $\mu$)
assuming the expansion history is given, and only the neutrino mass is free to
vary. Given the tight constraints on the neutrino mass, and for the profiles we
considered in this work, we recover numerically that the effect of such massive
neutrinos does not alter our conclusions. Finally, we stress that incorrectly
assuming a $\Lambda$CDM expansion history leads to a degraded reconstruction of
$\mu$, and/or a non-negligible bias in the
($\Omega_\mathrm{m,0}$,$\sigma_{8,0}$)-plane.
