The Grassmannian–Pfaffian categorical HP-duality conjecture

Let $W$ be a vector space of dimension $m$, and let $\operatorname{Pf}(2k,W)$ denote the closure of the locus of bivectors in $\Lambda^2W$ of rank $2k$. Let $\operatorname{Gr}(2,W)=\operatorname{Pf}(2,W)$ be the Grassmannian in its Plücker embedding, and let $W^\vee$ be the dual vector space. Grassmannian–Pfaffian HP-duality conjecture. For any $k$ there is a homological-projective duality between appropriate minimal categorical resolutions of $\operatorname{Pf}(2k,W)$ and $\operatorname{Pf}(2(\lfloor m/2\rfloor-k),W^\vee)$. When $m=\dim W$ is odd, these resolutions are strongly crepant. This proposes a higher-Pfaffian extension of Grassmannian–Pfaffian duality; the source gives no resolution.