Weil's global Torelli conjectures for K3 surfaces

Let $M$ be a fixed differentiable manifold underlying a K3 surface, and let

be complex structures on $M$ with holomorphic two-forms

and $'$. Write $[]$ and $[']$ for their periods in ${\mathbb P}(H^2(M,{\mathbb C}))$, and let ${\rm Diff}(M)$ be the diffeomorphism group of $M$, with identity component ${\rm Diff_o}(M)$. Let ${\rm O}(H^2(M,{\mathbb Z}))$ denote the group of lattice isomorphisms of $H^2(M,{\mathbb Z})$. Weil's global Torelli conjectures. i) If ${\mathbb C}[\sigma]={\mathbb C}[\sigma']\subset H^2(M,{\mathbb C})$, then there exists $f\in{\rm Diff_o}(M)$ such that $\sigma=f^*\sigma'$. ii) If $[\sigma]=g([\sigma'])$ for a lattice isomorphism $g\in{\rm O}(H^2(M,{\mathbb Z}))$, then there exists $f\in{\rm Diff}(M)$ such that $\sigma=f^*\sigma'$. The second version is established by the classical Global Torelli Theorem, whereas the first remains open as stated; the source itself notes that it requires additional qualifications to have a chance of being true.