  Belinskaya's theorem states that given an ergodic measure-preserving
transformation, any other transformation with the same orbits and an
$\mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows that
this theorem is optimal: for all $p<1$ the integrability condition on the
cocycle cannot be relaxed to being in $\mathrm{L}^p$. This also allows us to
answer a question of Kerr and Li: for ergodic measure-preserving
transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.
