My understanding is if there’s another minimum that juts down below the blue line in iteration 1 (top of part A), that part of the space is retained in iteration 2. It should keep the pseudosymmetric false minima across the iterations until the bound excludes them. Restated, eventually any non-marginalizing reconstruction will have to choose one alignment parameter; in cryoSPARC this should happen only after projection-matching against the current reconstruction gives error estimates that place the alternate parameters above the lower bound.
In a marginalizing program like Relion, you could instead end up with significant alignments at both minima and then have each particle inserted e.g. 40% at one and 60% at the other. Personally, I would rather have the single correct answer, but this approach can work well for some data (as we have no doubt all seen when using Relion).
The final option is something like pseudosymmetric refinement in Frealign (and the forthcoming cisTEM 2). There, the symmetry group is given with a lower-case code (“c4” instead of “C4”). Then the metadata file gets a line for each particle at each possible symmetry-related pose, each with its own occupancy and score. These scores are used for per-particle b-factor weighting, so particles can contribute low-frequency information to all symmetry-related poses and more high-frequency information just to the higher scoring one.
Assuming my understanding about cryoSPARC above is correct, then a possible tweak would be to always retain the poses which are symmetry-related to the best pose when computing the true error. This could go along well with limiting the search space for symmetric refinements, which cryoSPARC hasn’t done previously. It could then be requested using a checkbox for pseudosymmetric refinement, similar to giving a lower-case point group in Frealign.