The K3-type category conjecture for Küchle's fivefold hyperplane section

Let $W$ be a $7$-dimensional vector space, let $X\subset\operatorname{Gr}(3,W)$ be the $5$-dimensional zero locus of a section of $\Lambda^2\mathcal{U}_3^\vee\oplus\Lambda^3(W/\mathcal{U}_3)$, and let $Y$ be a hyperplane section of $X$. A rectangular Lefschetz decomposition is a semiorthogonal decomposition of the form $D^b\operatorname{Coh}(X)=\langle\mathcal{B},\mathcal{B}(1)\rangle$. K3-type category conjecture. The variety $X$ has a rectangular Lefschetz decomposition $D^b\operatorname{Coh}(X)=\langle\mathcal{B},\mathcal{B}(1)\rangle$ with $\mathcal{B}$ generated by $6$ exceptional objects. Consequently, $Y$ has a semiorthogonal decomposition $D^b\operatorname{Coh}(Y)=\langle\mathcal{A}_Y,\mathcal{B}\rangle$ with $\mathcal{A}_Y$ a category of K3 type. The claim predicts a K3-type residual category for this Küchle hyperplane section; the source gives no resolution.