Breuil–Schneider style functor for prismatic geometrization

Let $\mathrm{Stack}_{\mathrm{Bun},G,X,n}$ be the stack of $G$-bundles used in the paper, and let

$$\mathrm{QuasiCohSheaves}^{\blacksquare}_{\mathrm{prismatic},\mathrm{Stack}_{\mathrm{prismatic}}}(\mathrm{Stack}_{\mathrm{Bun},G,X,n})_{\mathrm{End}}$$

and

$$\mathrm{QuasiCohSheaves}^{\blacksquare}_{\mathrm{prismatic},\mathrm{Stack}_{\mathrm{prismatic}}}(\mathrm{Stack}_{\mathrm{Bun},G,X,n})_{\mathrm{Loc}}$$

be the indicated endomorphism-line and local subcategories. Breuil–Schneider style functor conjecture. The functor

$$\begin{aligned} &\otimes^{\blacksquare}:\mathrm{QuasiCohSheaves}^{\blacksquare}_{\mathrm{prismatic},\mathrm{Stack}_{\mathrm{prismatic}}}(\mathrm{Stack}_{\mathrm{Bun},G,X,n})_{\mathrm{End}}\\ &\times\mathrm{QuasiCohSheaves}^{\blacksquare}_{\mathrm{prismatic},\mathrm{Stack}_{\mathrm{prismatic}}}(\mathrm{Stack}_{\mathrm{Bun},G,X,n})_{\mathrm{Loc}}\\ &\rightarrow\mathrm{QuasiCohSheaves}^{\blacksquare}_{\mathrm{prismatic},\mathrm{Stack}_{\mathrm{prismatic}}}(\mathrm{Stack}_{\mathrm{Bun},G,X,n}) \end{aligned}$$

can be used to construct a geometrization of the Breuil–Schneider parametrization. The source presents this as a proposed construction rather than proving that it produces the claimed geometrization.