Voisin's strong Beauville–Voisin conjecture for hyper-Kähler varieties

Let $X$ be a complex hyper-Kähler variety, and let $A^*(X)=\bigoplus_i A^i(X)$ be its Chow ring with $\mathbb{Q}$-coefficients. A subvariety $Z\subset X$ is a constant cycle subvariety if all its points represent the same class in the Chow group of zero-cycles of $X$, and it is Lagrangian if its dimension equals its codimension. Write $\langle A^1(X),c_j(T_X),Z_k\rangle\subset A^*(X)$ for the $\mathbb{Q}$-algebra generated by divisors, the Chern classes $c_j(T_X)$ of the tangent bundle, and Lagrangian constant cycle subvarieties $Z_k\subset X$. Voisin's strong conjecture. The algebra

$$\langle A^1(X),c_j(T_X),Z_k\rangle\subset A^*(X)$$

should inject into cohomology under the cycle class map. The paper studies this strong form and verifies it in codimension larger than two for Hilbert squares of K3 surfaces, Fano varieties of lines in cubic fourfolds, and double EPW sextics; the general statement remains open.