We compare state-of-the-art semi-analytic models of galaxy formation as well as advanced subhalo abundance matching models with a large sample of early-type galaxies from Sloan Digital Sky Survey at z < 0.3. We focus our attention on the dependence of median sizes of central galaxies on host halo mass. The data do not show any difference in the structural properties of early-type galaxies with environment, at fixed stellar mass. All hierarchical models considered in this work instead tend to predict a moderate to strong environmental dependence, with the median size increasing by a factor of ̃1.5-3 when moving from low- to high-mass host haloes. At face value the discrepancy with the data is highly significant, especially at the cluster scale, for haloes above log Mhalo ≳ 14. The convolution with (correlated) observational errors reduces some of the tension. Despite the observational uncertainties, the data tend to disfavour hierarchical models characterized by a relevant contribution of disc instabilities to the formation of spheroids, strong gas dissipation in (major) mergers, short dynamical friction time-scales and very short quenching time-scales in infalling satellites. We also discuss a variety of additional related issues, such as the slope and scatter in the local size-stellar mass relation, the fraction of gas in local early-type galaxies and the general predictions on satellite galaxies.
