Can the adaptive metropolis algorithm collapse without the covariance lower bound?

The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix at step n+1 Sn = Cov(X1,..., Xn)+εI, that is, the sample covariance matrix of the history of the chain plus a (small) consta...