Variation on a theme of Beauville-Voisin

The Beauville-Voisin conjecture is about the Chow ring of hyper-Kähler varieties. We consider a variant version taking place in the ring of algebraic cycles modulo algebraic equivalence. We prove this variant version is true in codimension at least 3 for Fano varieties of lines on cubic fourfolds, and for double EPW sextics.