Consistency of Modified Gravity with a decreasing $G_{\rm eff}(z)$ in a
  $\Lambda$CDM background
  Recent analyses \cite{Nesseris:2017vor,Kazantzidis:2018rnb} have indicated
that an effective Newton's constant $G_{\rm eff}(z)$ decreasing with redshift
may relieve the observed tension between the Planck15 best fit $\Lambda$CDM
cosmological background ({\it i.e.} Planck15/$\Lambda$CDM) and the
corresponding $\Lambda$CDM background favored by growth $f\sigma_8$ and weak
lensing data. We investigate the consistency of such a decreasing $G_{\rm
eff}(z)$ with some viable scalar-tensor models and $f(R)$ theories. We stress
that $f(R)$ theories generically can not lead to a decreasing $G_{\rm eff}(z)$
for any cosmological background. For scalar-tensor models we deduce that in the
context of a $\Lambda$CDM cosmological background, a decreasing $G_{\rm
eff}(z)$ is not consistent with a large Brans-Dicke parameter $\omega_{BD,0}$
today. This inconsistency remains and amplifies in the presence of a phantom
dark energy equation of state parameter ($w < -1$). However it can be avoided
for $w >-1$. We also find that any modified gravity model with the required
decreasing $G_{\rm eff}(z)$ and $G_{{\rm eff},0}=G$, would have a
characteristic signature in its growth index $\gamma$ with $0.61\lesssim
\gamma_0\lesssim 0.69$ and large slopes $\gamma_0'$, $0.16\lesssim
\gamma_0'\lesssim 0.4$, which is a characteristic signature of a decreasing
(with $z$) $G_{\rm eff}(z)<G$ on small redshifts. This is a substantial
departure today from the quasi-static behaviour in $\Lambda$CDM with
$(\gamma_0,\gamma_0')\approx (0.55,-0.02)$.
