Kuznetsov's homological projective duality conjecture for Pfaffian varieties

Let $W$ be a vector space of dimension $2n$. For $1\le k\le n-1$, let $\operatorname{Pf}_k(W^*)\subset\mathbb{P}(\Lambda^2W^*)$ be the variety of skew-forms of rank at most $2n-2k$, and let $\widetilde{\operatorname{Pf}}_k(W^*)$ denote a categorical resolution of singularities. Kuznetsov's conjecture. The Pfaffian varieties $\operatorname{Pf}_k(W^*)$ admit categorical resolutions of singularities $\widetilde{\operatorname{Pf}}_k(W^*)$ such that

$$\widetilde{\operatorname{Pf}}_k(W^*)$$

is Homologically Projectively Dual to $\widetilde{\operatorname{Pf}}_{n-1-k}(W)$. In particular, the Grassmannian of lines $\operatorname{Gr}(2,W)$ is Homologically Projectively Dual to a certain categorical resolution of singularities $\widetilde{\operatorname{Pf}}(W^*)$ of the Pfaffian hypersurface $\operatorname{Pf}(W^*)$. This extends the classical projective-duality relation $(\operatorname{Pf}_k(W^*))^\vee=\operatorname{Pf}_{n-1-k}(W)$; the conjecture was proved for $n=3$, while the general statement is not resolved in the supplied source.