Lefschetz decomposition conjecture for the congruence of lines and Peskine variety

Let $\mathsf{A}$ be the K3 category of the Debarre–Voisin hyperplane section $Y$, and let $T$ and $P$ be the associated congruence of lines and Peskine variety. Let $\mathcal{U}$ and $\mathcal{Q}$ denote the bundles used in the construction, and write $(1)$ for tensoring with the relevant polarization. Improved Lefschetz decomposition conjecture. There should be a fully faithful functor $\Phi:\mathsf{A}\to\operatorname{D}^b(T)$ such that

$$\mathsf{B}=\langle\Phi(\mathsf{A}),\mathcal{O},\mathcal{U}^*,S^2\mathcal{U}^*\rangle\subset\operatorname{D}^b(T)$$

provides

$$\operatorname{D}^b(T)=\langle\mathsf{B},\mathsf{B}(1),\mathsf{B}(2)\rangle.$$

There should also be a fully faithful functor $\Psi:\mathsf{A}\to\operatorname{D}^b(P)$ such that

$$\mathsf{C}_1=\langle\Psi(\mathsf{A}),\mathcal{O}\rangle\subset\mathsf{C}_0=\langle\Psi(\mathsf{A}),\mathcal{O},\mathcal{Q}\rangle\subset\operatorname{D}^b(P)$$

provides

$$\operatorname{D}^b(P)=\langle\mathsf{C}_0,\mathsf{C}_1(1),\mathsf{C}_1(2)\rangle.$$

These decompositions refine the expected three-copy K3 structures and use the exceptional collections constructed for $T$ and $P$; the source gives no proof.