- by setting the inverse noise to zero in the cut
- by including a foreground component with infinite prior variance in the cut and adding a new Gibbs step

Question: what did the authors actually choose for the noise N_cut in the cut?

It could be chosen to be the experimental noise, but to some extent this seems to be an arbitary choice (unless perhaps you are interested in learning about foregrounds). In the limit N_cut = 0, the Gibbs iterations never converge. In the limit N_cut is inifinite, (2) becomes the same as (1). Somewhere in between, there may be an optimal choice. Having N_cut the experimental noise (or average of the out-of-cut noise) probably optimises the matrix inversion, but it's not clear this is the best choice for overall performance if it makes for slow Gibbs convergence. So what should you choose for N_cut?