Convergence of constraints of the reionization optical depth as a function of the number of main chain samples used to construct the CMB mean map and covariance matrix and the relative wall time needed to produce such samples in the main Gibbs loop. The solid blue line shows the posterior mean for τ, while the gray and green regions show the corresponding 68% confidence interval for Natale et al. (2020) and Tristram et al. (2022), respectively.