There is a functor from the category of higher displays over $k$ of type $\mu$ to the category of isocrystals over $k$ where $k$ is an algebraically closed field where you forget about the filtration and divided Frobenius.Lau generalizes this concept and defined $(G,\mu)$ displays and again you get a functor to isocrystal with $G$ structures $B(G)$.
I want to understand the image of this functor. I think this functor factors through crystals with Hodge polygon $\mu$, and the image consists of the isocrystals that come from a crystal with Hodge polygon $\mu$, And I think this is the set that people call $B(G,\mu)$(or $B(G,\mu^{-1}$)!)
I want to know is my understanding of the image is correct and is there a reference for this image in literature at least in the case $G=GL_n$?
