Derived equivalence conjecture for Azumaya algebras on determinantal Calabi–Yau threefolds

Let $\widehat{X}$ be a small resolution of $X_{\vec{d}}$, let $S_{\vec{d},1}$ be the relevant singular locus, and let $f:\widehat{X}\to\mathbb{P}^3$ be the morphism in the construction. Let $\mathcal{B}$ and $\mathcal{B}_0$ be the sheaves of algebras appearing in the paper. Derived equivalence conjecture. There should exist an Azumaya algebra $\widehat{\mathcal{B}}$ on $\widehat{X}$ extending $\mathcal{B}|_{X_{\vec{d}}-S_{\vec{d},1}}$, with $f_*\widehat{\mathcal{B}}=\mathcal{B}_0$, such that the two displayed functors in the source are inverse derived equivalences. Moreover, for every exceptional curve $C_p$,

$$\widehat{\mathcal{B}}|_{C_p}\simeq \underline{\operatorname{End}}\left(\left(\mathcal{O}_{C_p}\oplus\mathcal{O}_{C_p}(-1)\right)^{\oplus 2^{n-2}}\right).$$

This conjecture extends the known $k=4$ case and is intended to relate torsion-refined invariants to algebraic sheaves of $\mathcal{B}_0$-modules; the general statement is not proved in the source.