Homological projective duality conjecture for generalized Pfaffian varieties

Let $W$ be a vector space of dimension $n$. For each integer $t$ with

$$0\leq t\leq \lfloor n/2\rfloor,$$

consider the generalized Pfaffian varieties

$$X = \operatorname{Pf}(2t,W),\qquad Y = \operatorname{Pf}(2\lfloor n/2\rfloor - 2t,W^*).$$

A noncommutative resolution of singularities is a resolution of the corresponding variety by a noncommutative algebra.

Generalized Pfaffian HPD conjecture. For every $0\leq t\leq \lfloor n/2\rfloor$, there exist noncommutative resolutions of singularities of $X$ and $Y$ which are Homologically Projectively Dual.

This would extend homological projective duality beyond the basic Grassmannian–Pfaffian pair to generalized Pfaffian varieties. It is proposed as a possible direction of generalization, and no proof or resolution is given in the paper.