In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived categories of) smooth representations of $G:=\mathrm{GL}_{2}(F)$ on $\mathcal{O}$-modules, for some finite extension $F$ of $\mathbb{Q}$ and some coefficient ring $\mathcal{O}$ (say the ring of integers of some other finite extension $L$ of $\mathbb{Q}_{p}$) to coherent sheaves on the Emerton-Gee-Stack, with certain properties which I will not elaborate here.

This functor is supposed to take the form $\pi\mapsto L_{\infty}\otimes_{\mathcal{O}[[G]]}\pi$ (remark 6.1.23 of the article), where $L_{\infty}$ is a sheaf on the Emerton-Gee stack that is supposed to be related to Taylor-Wiles patching. Namely, Galois deformation rings are supposed to be versal rings of the Emerton-Gee stack, and pulling back $L_{\infty}$ to the Galois deformation ring should give us the usual patched module $M_{\infty}$.

The same article also formulates a global conjecture (expected theorem 9.4.2, itself a special case of conjecture 9.3.2). The global conjecture expresses (a localization w.r.t. a certain maximal ideal) the (completed) cohomology of a Shimura variety as the global sections of a sheaf on the stack of *global* Galois representations, which is the tensor product of the pullback of $L_{\infty}$ through the "localization" map to the Emerton-Gee stack, with the vector bundle corresponding to the universal global Galois representation.

On the other hand, the p-adic local Langlands correspondence has also been realized using the p-adic cohomology of the Drinfeld tower, in the work of Colmez-Dospinescu-Niziol. There is also related work by Scholze. This work is not discussed in the Emerton-Gee-Hellmann article (as they point out in their introduction).

**So my question is, does the p-adic cohomology of the Drinfeld/Lubin-Tate tower fit into the framework of Emerton-Gee-Hellmann, and how?**

Their local conjecture is speculated to be able to be upgraded to an equivalence of categories similar to conjecture X.1.4 of Fargues-Scholze, and the functor of Emerton-Gee-Hellmann is speculated to be induced by the inclusion $[*/G(F)]\hookrightarrow \mathrm{Bun}_{G}$. In the setting of Fargues-Scholze, the Drinfeld/Lubin-Tate tower appears as a fiber of the Hecke stack above a point of $\mathrm{Bun}_G$. Currently, the setting of Fargues-Scholze cannot yet be transported to the p-adic case, and so the approach of Emerton-Gee-Hellmann (or any other approaches for that matter for p-adic local Langlands) do not involve $\mathrm{Bun}_{G}$ or the Hecke stack.

Still, the work of Colmez-Dospinescu-Niziol, which builds on the prior work of Dospinescu-Le Bras, as well as the previously mentioned work of Scholze on the p-adic cohomology of the Lubin-Tate tower, involve Emerton's local-global compatibility with completed cohomology, which appears to underlie the Emerton-Gee-Hellmann framework. So it does seem like it should fit into the framework somehow, but I do not know how.