  Models of dark energy or modified gravity that tries to alleviate the
tensions on the Hubble constant ($H_0$) and the matter fluctuation parameter
($\sigma_8$) are usually parameterized as function of either late or early time
cosmic evolution. In this work we rather focus on one that could privilege
extensions to $\Lambda$CDM on intermediate redshifts by mean of a Gaussian-like
window function with a free moving centre $a_{Gwin}$ combined with a modified
gravity parameter $\mu_{Gwin}$ and an extension of the equation of state
parameter $\omega_{Gwin}$. Using different combinations of the latest available
current datasets subject of the discrepancies, such as the cosmic microwave
(CMB) background power spectrum, the baryonic acoustic scale (BAO) in galaxy
distribution, Weak lensing (WL) shear and galaxy clustering cross correlations
and local hubble constant measurements, we investigate whether such model could
alleviate each or both $H_0$ and $\sigma_8$ tensions. We found when combining
all probes that the $\sigma_8$ tension is alleviated while the $H_0$ is reduced
with a small preference for a positive $\omega_{Gwin}$ without a particular
preference for a redshift or a $\mu_{Gwin}$ different from its equivalent
$\Lambda$CDM value. However, if we follow another approach and compare the two
sets of the probes subject of discrepancy i.e. CMB+BAO vs WL+local $H_0$, we
found that the model is able of solving the $\sigma_8$ discrepancy at the
expense of a enlargement of the constraints, while the Hubble constant
discrepancy is not that affected due to the fact that the two likelihood
contours are stretched in parallel directions. We conclude that modifying
$\Lambda$CDM cosmology at intermediate redshifts within our model, and the
constraints from the datasets used in this study, are not likely a viable
solution to solve both tensions.
