Semiorthogonal decomposition conjecture for Pfaffian linear sections

Let $W$ be a vector space of dimension $2n$, let $V\subset\Lambda^2W^*$ satisfy $\dim V=2n=\dim W$, and let $V^\perp\subset\Lambda^2W$ be its orthogonal. Define the Pfaffian hypersurface linear section

$$Y_V=\mathbb{P}(V)\cap\operatorname{Pf}(W^*)$$

and the dual Grassmannian linear section

$$X_V=\mathbb{P}(V^\perp)\cap\operatorname{Gr}(2,W).$$

Let $\widetilde{Y}_V$ be a categorical resolution of singularities of $Y_V$. Pfaffian linear-section decomposition conjecture. The derived category of coherent sheaves on $\widetilde{Y}_V$ has a semiorthogonal decomposition whose nontrivial component is equivalent to the derived category of $X_V$; more precisely,

$$\mathcal{D}^b(\widetilde{Y}_V)=\langle \mathcal{D}^b(X_V),\mathcal{O},\mathcal{O}(1),\dots,\mathcal{O}(n-1)\rangle.$$

This is predicted by homological projective duality for the Grassmannian and Pfaffian hypersurface. In the supplied text it is explicitly deduced from the preceding conjecture, and no independent resolution status is given.