Debarre–Voisin realization conjecture for Hilbert squares of genus 16 K3 surfaces

Let $(S,h)$ be a general polarized K3 surface of genus $16$, let $S'$ be its unique nontrivial Fourier–Mukai partner, and let $V_{10}$ and $V_{10}'$ be the rank $10$ vector spaces constructed from $S$ and $S'$, respectively. Let $t_1\in\wedge^3 V_{10}^*$ be the trivector corresponding to the trivector $t_2'\in\wedge^3 V_{10}'$ constructed from $S'$. For a trivector $t_1\in\wedge^3 V_{10}^*$, write $DV(t_1)\subset G(6,V_{10})$ for the associated Debarre–Voisin variety. Debarre–Voisin realization conjecture. There exists a canonical isomorphism

$$S^{[2]}\cong DV(t_1)\subset G(6,V_{10}).$$

The conjecture proposes a canonical Debarre–Voisin description of the Hilbert square of a general genus $16$ K3 surface; the supplied text says it has been verified for some examples by computer, but does not state a general proof.