Equivalence of stacks of twisted differential-module categories

A stack of twisted $\mathbb{C}_X$-modules is a stack whose objects are modules over the sheaf of differential operators $\mathcal{D}_X$ with coefficients twisted by $\mathbb{C}_X$. Let $\mathfrak{M}$ be a stack of twisted $\mathbb{C}_X$-modules, and write $\mathfrak{Mod}(\mathcal{D}_X;\mathfrak{M})$ for the corresponding stack of twisted $\mathcal{D}_X$-modules.

Equivalence conjecture. Any stack of $\mathbb{C}_X$-twisted $\mathcal{D}_X$-modules is $\mathbb{C}$-equivalent to a stack of the form

$$\mathfrak{Mod}(\mathcal{D}_X;\mathfrak{M})$$

for some stack of twisted $\mathbb{C}_X$-modules $\mathfrak{M}$.

This claim concerns the classification of stacks of twisted differential-module categories by stacks of twisted coefficient modules. The supplied text does not state whether the claim has been proved or disproved.