  We explain a strategy for distinguishing Brill-Noether loci in the moduli
space of curves by studying the lifting of linear systems on curves in
polarized K3 surfaces, which motivates a conjecture identifying the maximal
Brill-Noether loci with respect to containment. Via an analysis of the
stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear
systems of rank 3 which suffice to prove the maximal Brill-Noether loci
conjecture in genus 9-19, 22, and 23.
