Before revision:  The counterexamples for CoVaR based on the stress event X=VaR_\alpha(X) include inconsistency with respect to correlation in the bivariate Gaussian model .
After revision:  The counterexamples for CoVaR based on the stress event X=VaR_\alpha(X) include inconsistency with respect to correlation in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X>=VaR_\alpha(X) is an increasing function of the dependence parameter .