Schapira's reflexive extension conjecture for DQ-modules

Let $(X,\mathcal{A}_X)$ be a complex manifold endowed with a DQ-algebroid $\mathcal{A}_X$. A coherent $\mathcal{A}_X$-module $\mathcal{M}$ is $d$-reflexive if it has no $\hbar$-torsion, $\mathcal{E}xt_{\mathcal{A}_X}^j(\mathcal{M},\mathcal{A}_X)=0$ for $j<d$, and the natural biduality morphism is an isomorphism in degree $d$. Let $Y$ be a closed complex analytic subset of $X$ with $\operatorname{codim}_X Y\geq d+3$, and let $j:X\setminus Y\hookrightarrow X$ be the open inclusion. Let $\mathcal{M}$ be a coherent $\mathcal{A}_X|_{X\setminus Y}$-module.

Schapira's conjecture. If $\mathcal{M}$ is $d$-reflexive, then $j_*\mathcal{M}$ is a coherent $\mathcal{A}_X$-module.

This extends the codimension-three conjecture from holonomic DQ-modules on symplectic manifolds to DQ-modules on arbitrary complex Poisson manifolds. The paper explains that the existing proof does not adapt directly because general Poisson manifolds do not locally have a unique star-algebra; the conjecture remains open.