Left panel: relative amplitudes of the prior term (solid lines; ) and inverse-noise term (diffuse bands; ) on the diagonal of the coefficient matrix A. Each full inverse-noise band has been evaluated by summing over the nine Planck frequency bands. The darker part of each band displays the extents of the inverse-noise terms if one reduces contrast by adding synthetic regularization noise to the 1% least noisy pixels. This in turn reduces the condition number of the full matrix. Where the solid lines lie above the diffuse bands, the full matrix is prior dominated; where they lie below, the full matrix is data dominated. Right panel: schematic setup of which term dominates A in different regimes. The pseudo-inverse preconditioner of Sect. 3 automatically resolves the low-ℓ versus high-ℓ split, taking into account cross-component couplings. To handle the crucial low-ℓ regime inside the mask (red), the extension described in Sect. 4 is required.