Before revision:  The second method is parametric : solve the empirical Markowitz problem instead if the log-return distribution is in the elliptical family (which includes Gaussian and t distributions), as then the population frontiers of the Markowitz and mean-CVaR problems are equivalent . Numerical simulations show both methods improve upon the empirical mean-CVaR solution under an elliptical model, with the Markowitz solution dominating.
After revision:  The second method is parametric : solve the empirical Markowitz problem instead if the log-return distribution is in the elliptical family (which includes Gaussian and t distributions), as then the population frontiers of the Markowitz and mean-CVaR problems are equivalent when the log-return distribution is elliptical. We derive the asymptotic behavior of the nonparametric PBR solution, which leads to insight into the effect of penalization, and justification of the parametric PBR method. We also show via simulations that the PBR methods produce efficient frontiers that are, on average, closer to the population efficient frontier than the empirical approach to the mean-CVaR solution under an elliptical model, with the Markowitz solution dominating.