  Let $X$ be a hyperk\"ahler variety, and let $Z\subset X$ be a Lagrangian
subvariety. Conjecturally, $Z$ should have trivial intersection with certain
parts of the Chow ring of $X$. We prove this conjecture for certain Hilbert
schemes $X$ having a Lagrangian fibration, and $Z\subset X$ a general fibre of
the Lagrangian fibration.
