Laumon's conjecture on loose Hecke eigensheaves

Let $G$ be a reductive group and let $X$ be a smooth proper curve. A loose Hecke eigensheaf is an object $\CF\in \Dmod(\Bun_G)$ such that for every $V\in \Rep(\cG)$ there are an object $E_V\in \Dmod_{\on{lisse}}(X)$ and an object $\CF'\in \Dmod(\Bun_G)$ with

$$H(V,\CF)\simeq \CF'\otimes E_V.$$

Laumon's conjecture on loose Hecke eigensheaves. A loose Hecke eigensheaf has a nilpotent singular support. This is presented as a special case of the preceding inclusion conjecture and as a version of a conjecture first proposed by G. Laumon for actual Hecke eigensheaves; the source does not state that it is resolved.