The categorical crepant resolution conjecture for Pfaffian varieties

Let $V$ be an $n$-dimensional vector space, with $n$ even, and let $k$ be an integer. Write ${\mathrm {Pf}}(2k,V^{\vee})$ and ${\mathrm {Pf}}(n-2k,V)$ for the corresponding Pfaffian varieties, and let $\widetilde{D}^b$ denote their categorical crepant resolutions. A Lefschetz decomposition consists of admissible blocks $\mathcal{A}_i$ or $\mathcal{B}_i$ with the indicated twists.

Categorical crepant resolution conjecture. The categorical crepant resolution $\widetilde{D}^b({\mathrm {Pf}}(2k,V^{\vee}))$ has the non-rectangular Lefschetz decomposition

$$\widetilde{D}^b({\mathrm {Pf}}(2k,V^{\vee}))=\Big\langle \mathcal{A}_{0}, \mathcal{A}_{1}(1),\ \cdots,\mathcal{A}_{nk-\frac{n}{2}-1}(nk-\frac{n}{2}-1),\ \mathcal{A}_{nk-\frac{n}{2}}(nk-\frac{n}{2}),\ \cdots,\mathcal{A}_{nk-1}(nk-1) \Big\rangle,$$

where $\mathcal{A}_0=\cdots=\mathcal{A}_{nk-\frac{n}{2}-1}$ are $nk-\frac{n}{2}$ blocks of size $\binom{n/2}{k}$, and $\mathcal{A}_{nk-\frac{n}{2}}=\cdots=\mathcal{A}_{nk-1}$ are $n/2$ blocks of size $\binom{n/2-1}{k}$. Similarly, the dual Lefschetz decomposition of $\widetilde{D}^b({\mathrm {Pf}}(n-2k,V))$ is

$$\widetilde{D}^b({\mathrm {Pf}}(n-2k,V))=\Big\langle \mathcal{B}_{\frac{n^2}{2}-nk-1}(-\frac{n^2}{2}+nk+1),\ \cdots \mathcal{B}_{\frac{n(n-1)}{2}-nk}(-\frac{n(n-1)}{2}+nk),\ \mathcal{B}_{\frac{n(n-1)}{2}-nk-1}(-\frac{n(n-1)}{2}+nk+1),\ \cdots,\mathcal{B}_{1}(-1),\mathcal{B}_0\Big\rangle,$$

where $\mathcal{B}_0=\cdots=\mathcal{B}_{\frac{n(n-1)}{2}-nk-1}$ are $\frac{n(n-1)}{2}-nk$ blocks of size $\binom{n/2}{n/2-k}$, and $\mathcal{B}_{\frac{n(n-1)}{2}-nk}=\cdots=\mathcal{B}_{\frac{n^2}{2}-nk-1}$ are $n/2$ blocks of size $\binom{n/2-1}{n/2-k}$. This predicts the Lefschetz structures required for Homological Projective Duality in the even-dimensional Pfaffian case, which remains open because the relevant categorical crepant resolutions and decompositions have not yet been constructed.