An extensive empirical study of the class of Volterra Bergomi models using SPX options data between 2011 and 2022 reveals the following fact-check on two fundamental claims echoed in the rough volatility literature: Do rough volatility models with Hurst index H ∈ (0, 1/2) really capture well SPX implied volatility surface with very few parameters? No, rough volatility models are inconsistent with the global shape of SPX smiles. They suffer from severe structural limitations imposed by the roughness component, with the Hurst parameter H ∈ (0, 1/2) controlling the smile in a poor way. In particular, the SPX at-the-money skew is incompatible with the power-law shape generated by rough volatility models. The skew of rough volatility models increases too fast on the short end, and decays too slow on the longer end where "negative" H is sometimes needed. Do rough volatility models really outperform consistently their classical Markovian counterparts? No, for short maturities they underperform their one-factor Markovian counterpart with the same number of parameters. For longer maturities, they do not systematically outperform the one-factor model and significantly underperform when compared to an under-parametrized two-factor Markovian model with only one additional calibratable parameter. On the positive side: our study identifies a (non-rough) path-dependent Bergomi model and an under-parametrized two-factor Markovian Bergomi model that consistently outperform their rough counterpart in capturing SPX smiles between one week and three years with only 3 to 4 calibratable parameters.