also, i still think it should be relatively easy to extend the kmeans++ to the gmm++ (perhaps in the fully constrained setting), to show that the algorithm (rather than the estimator) is asymptotically unbiased. that is stronger than the estimator being unbiased, because there might not, in general, be a polynomial time algorithm that converges.
based on kmeans++
also, i still think it should be relatively easy to extend the kmeans++ to the gmm++ (perhaps in the fully constrained setting), to show that the algorithm (rather than the estimator) is asymptotically unbiased. that is stronger than the estimator being unbiased, because there might not, in general, be a polynomial time algorithm that converges.