Homological projective duality conjecture for the Pfaffian-Grassmannian pair

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A Lefschetz decomposition is a semi-orthogonal decomposition of the derived category into blocks with successive twists. Homological projective duality conjecture. There exists a Lefschetz decomposition of $\mathbf{D}_{\mathsf{coh}}^b(\mathbf{G})$ and a smooth variety $\overline{\mathbf{Pf}}\to\mathbf{P}^*$, possibly understood in an extended sense such as a non-commutative scheme, mapping to $\mathbf{Pf}$ and homologically projectively dual to $\mathbf{G}$ with respect to this decomposition. Furthermore, Theorem~ is a direct application of the main result of~. The conjecture proposes the homological-projective-dual geometric object needed to explain the derived equivalences between smooth linear sections of the Grassmannian and Pfaffian varieties; the supplied text gives no evidence that this formulation has been resolved.