Full faithfulness of the extended Whittaker coefficient functor

Let $G$ be a connected reductive group with connected center. Write $\type{\coeff}_{G,\ext}: \Dmod(\Bun_G)\to\Wh(G,\ext)$ for the extended Whittaker coefficient functor from $D$-modules on the moduli stack of $G$-bundles to the extended Whittaker category. Extended Whittaker full-faithfulness conjecture. The functor

$$\type{\coeff}_{G,\ext}: \Dmod(\Bun_G)\to\Wh(G,\ext)$$

is fully faithful. This conjecture extends the proved result for $G=GL_n$ and $G=PGL_n$; the source notes that its classical analogue is false in general, so the conjecture is refuted.