Projective normality and quadratic generation for polarized Hilbert squares of K3 surfaces

Let $(S,L)$ be a polarized $K3$ surface of genus $g$ with $\operatorname{Pic}(S)=\mathbb{Z}\cdot L$. Consider the embedding defined by the complete linear system $|L_2-2\delta|$:

$$S^{[2]}\hookrightarrow \mathbb{P}\left(H^0(S^{[2]},L_2-2\delta)^\vee\right).$$

Projective-normality and quadratic-generation conjecture. If $g\geq8$ (equivalently, $q(L_2-2\delta)\geq6$), then this embedding is projectively normal; and if $g\geq10$ (equivalently, $q(L_2-2\delta)\geq10$), then the homogeneous ideal of $S^{[2]}$ is generated by quadrics.

These expectations concern the projective geometry of Hilbert squares in arbitrary genus and are motivated by the authors' results in genera $7$ and $8$, together with computer experiments. The source presents them as reasonable expectations rather than established results, so they remain open on the supplied evidence.