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Portfolio selection based on high-dimensional covariance matrices is a key challenge in data-rich environments with the curse of dimensionality severely affecting most of the available covariance models. We challenge several multivariate Dynamic Conditional Correlation (DCC)-type and Stochastic Volatility (SV)-type models to obtain minimum-variance and mean-variance portfolios with up to 1000 assets. We conclude that, in a realistic context in which transaction costs are taken into account, although DCC-type models lead to portfolios with lower variance, modeling the covariance matrices as latent Wishart processes with a shrinkage towards the diagonal covariance matrix delivers more stable optimal portfolios with lower turnover and higher information ratios. Our results reconcile previous findings in the portfolio selection literature as those claiming for equicorrelations, a smooth dynamic evolution of correlations or correlations close to zero.