Optimal experimental design (OED) concerns itself with identifying ideal methods of data collection, e.g. via sensor placement. The greedy algorithm, that is, placing one sensor at a time, in an iteratively optimal manner, stands as an extremely robust and easily executed algorithm for this purpose. However, it is a priori unclear whether this algorithm leads to sub-optimal regimes. Taking advantage of the author’s recent work on non-smooth convex optimality criteria for OED, we here present a framework for rejection of sub-optimal greedy indices, and study the numerical benefits this offers.