The codimension-three extension conjecture for microdifferential modules

Let $X$ be a complex manifold, let $Y\subset X$ be a subvariety, and write

$$j\colon X\setminus Y\hookrightarrow X$$

for the open embedding. Let $\mathcal{A}_X$ denote the relevant sheaf of microdifferential operators, and let $\mathcal{N}$ be a reflexive coherent $\mathcal{A}_{X\setminus Y}$-module. Codimension-three extension conjecture. If

$$\dim Y\leq \dim X-3,$$

then $j_*\mathcal{N}$ is a coherent $\mathcal{A}_X$-module. This is the microdifferential analogue of the classical codimension-three extension theorem; the paper presents it as the convergent case that would imply the corresponding extension result, but the supplied text gives no resolution status.