By analogy with Green's Conjecture on syzygies of canonical curves, the
Prym-Green conjecture predicts that the resolution of a general level p
paracanonical curve of genus g is natural. The Prym-Green Conjecture is known
to hold in odd genus for almost all levels. Probabilistic arguments strongly
suggested that the conjecture might fail for level 2 and genus 8 or 16. In this
paper, we present three geometric proofs of the surprising failure of the
Prym-Green Conjecture in genus 8, hoping that the methods introduced here will
shed light on all the exceptions to the Prym-Green Conjecture for genera with
high divisibility by 2.