Geometricity of low-energy sheaf quantizations

Let $X$ be a symplectic manifold, let $\bm L$ be a Lagrangian brane, and let $c_{\bm L}>0$ be as in the classification theorem. A sheaf quantization of $\bm L$ in $(X,u<c_{\bm L})$ is called geometric if it is in the image of the classification map

$$\mathrm{cl}\colon H^1(L,_0^+/T^{c_{\bm L}}_0)\to (\bm L,c_{\bm L}).$$

The geometricity conjecture. For any $X$ and any $\bm L$, every sheaf quantization in $(X,u<c_{\bm L})$ is geometric. The supplied text gives no result establishing this assertion.