$\mathcal{G}$-$\mu$-displays and local shtuka

Two approaches to the construction of integral models of local Shimura-varieties are compared: that of Bültel-Pappas using $\mathcal{G}$-$\mu$-displays and that of Scholze using local mixed-characteristic shtuka. As an application, the representability of the diamond of the generic fiber of the Bültel-Pappas moduli problem is verified and a local representability conjecture of Rapoport-Pappas is proven. The methods apply to all unramified local Shimura-data under a certain condition on slopes and if $p\mathrm{\ne }2;$ thus in particular also to exceptional groups.