Towards homological projective duality for $\mathrm{G}\mathrm{r}(2,2n)$

Consider a Grassmannian $\mathrm{G}\mathrm{r}(2,V)$ for an even-dimensional vector space $V$. Its derived category of coherent sheaves has a Lefschetz exceptional collection with respect to the Plücker embedding. We consider a variety $X_1$ of pairs consisting of a degenerate $2$-form on $V$ and a line in its kernel. Note that $X_1$ is generically a ${\mathbb{P}}^1$-fibration over the Pfaffian variety of degenerate $2$-forms on $V$. We construct an exceptional collection of coherent sheaves on $X_1$ such that the subcategory of $D_{\mathrm{c}\mathrm{o}\mathrm{h}}^b(X_1)$ generated by that collection is conjecturally equivalent to the homologically projectively dual category of the Grassmannian.