Is $\omega_0 \omega_a$CDM a good model for the clumpy Universe?

The DESI collaboration just obtained a set of precise BAO measurements, that combined with CMB and SNIa datasets show that the $\omega_0 \omega_a$CDM model is preferred over $\Lambda$CDM, at more than $4\,\sigma$, to describe the dynamics of the expanding Universe. This raises the question whether this model also suitably describes the clumpy Universe. Also lately, detailed analyses of diverse cosmic tracers resulted in a new dataset of measurements of an observable from the clumpy Universe: $\sigma_8(z)$, spanning a high-redshift data $z \in [0.013, 3.8]$. In this work we use this dataset of 15 $\sigma_8(z_i)$ measurements to study the viability of the $\omega_0 \omega_a$CDM cosmological model to explain the clustered Universe. Our analyses compare the $\omega_0 \omega_a$CDM model with the $\sigma_8(z)$ function reconstructed from the data points using Gaussian Process. Moreover, we perform a similar evaluation of the $\Lambda$CDM model considering Planck and~DESI best-fit parameters. In addition, we implemented robustness tests regarding Gaussian Process reconstruction to support our results.