  Consider a Grassmannian $\mathrm{Gr}(2, V)$ for an even-dimensional vector
space $V$. Its derived category of coherent sheaves has a Lefschetz exceptional
collection with respect to the Pl\"ucker 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
$D^b_{\mathrm{coh}}(X_1)$ generated by that collection is conjecturally
equivalent to the homologically projectively dual category of the Grassmannian.
