  The dark matter halo sparsity, i.e. the ratio between spherical halo masses
enclosing two different overdensities, provides a non-parametric proxy of the
halo mass distribution which has been shown to be a sensitive probe of the
cosmological imprint encoded in the mass profile of haloes hosting galaxy
clusters. Mass estimations at several overdensities would allow for multiple
sparsity measurements, that can potentially retrieve the entirety of the
cosmological information imprinted on the halo profile. Here, we investigate
the impact of multiple sparsity measurements on the cosmological model
parameter inference. For this purpose, we analyse N-body halo catalogues from
the Raygal and M2Csims simulations and evaluate the correlations among six
different sparsities from Spherical Overdensity halo masses at
$\Delta=200,500,1000$ and $2500$ (in units of the critical density).
Remarkably, sparsities associated to distinct halo mass shells are not highly
correlated. This is not the case for sparsities obtained using halo masses
estimated from the Navarro-Frenk-White (NFW) best-fit profile, that
artificially correlates different sparsities to order one. This implies that
there is additional information in the mass profile beyond the NFW
parametrization and that it can be exploited with multiple sparsities. In
particular, from a likelihood analysis of synthetic average sparsity data, we
show that cosmological parameter constraints significantly improve when
increasing the number of sparsity combinations, though the constraints saturate
beyond four sparsity estimates. We forecast constraints for the CHEX-MATE
cluster sample and find that systematic mass bias errors mildly impact the
parameter inference, though more studies are needed in this direction.
