Lagrangian subvarieties in the Chow ring of some hyperkähler varieties

Let $X$ be a hyperkähler 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.