the bottom panel of Fig. \[fig2\], where the cloud-shadow pairs with polydispersity $s=0.3$ are displayed for different cases of size-dependence. It can be clearly seen that while, for cases I and V the same trend is observed, case IV seems to suggest a *widening* of the phase coexistence region (and hence a favouring of the phase transition).

In view of the lack of numerical simulation for polydisperse SHS to compare with, we have no way, at the present stage, to check how realistic those results are. On the other hand, we might suspect, based on the comparison in the one-component case, $C_{0}$ to fail to provide accurate representation in the region of low temperatures and high densities. This is the reason why other possible approaches have been recently tested. We now illustrate a different perturbative approach, which has proved to be promising in this context.

Perturbative treatment of the SHS-PY solution
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The main difficulty in dealing with the PY solution for polydisperse SHS stems from the solution of the coupled quadratic system of equations ([eq4]{}), as remarked. As the one-component case has a well-defined solution, one might then suspect that – for weak polydispersity – a perturbative expansion around 