How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

Question

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.

Answer 1

Briefly (I will elaborate below): One expects that their fully faithful functor from (roughly) $p$-adic representations of $G(\mathbb Q_p)$ to (roughly) coherent sheaves on the Emerton--Gee stack extends to an equivalence between (roughly) $p$-adic sheaves on $\mathrm{Bun}_G$ and (roughly) coherent sheaves on the Emerton--Gee stack. The latter equivalence should be compatible with Hecke operators, and the cohomology of Lubin--Tate/Drinfeld space is an instance of a Hecke operator.

In slightly more detail: There is now this thing called "categorical local Langlands", that seeks to describe the category of smooth representations of $G(\mathbb Q_p)$ in terms of Langlands dual data. More precisely, as conjectured by Hellmann and partly proved by Ben-Zvi--Chen--Helm--Nadler, there should be a fully faithful functor from the derived category of representations of $G(\mathbb Q_p)$ towards $\mathrm{Ind}D^b_{\mathrm{coh}}$ of the stack of $L$-parameters. The conjectures of Emerton--Gee--Hellmann are $p$-adic variants of this.

On the other hand, the conjectures of Zhu and Fargues and myself predict that this fully faithful functor lifts to an equivalence between the derived category of $\overline{\mathbb Q}_\ell$-sheaves on $\mathrm{Bun}_G$, and all of $\mathrm{Ind}D^b_{\mathrm{coh}}$ of the stack of $L$-parameters. (One can also formulate a $\overline{\mathbb Z}_\ell$-statement.) The general expectation is that the conjectures of Emerton--Gee--Hellmann admit a similar lift, but this requires a good theory of $p$-adic sheaves on $\mathrm{Bun}_G$, and this is a nontrivial task. Mann's thesis should hopefully be helpful here, but in some sense his category is too big (and it's not obvious how to cut it down to the right size).

In my work with Fargues, we also explain that this conjectural equivalence should be compatible with Hecke operators, and that in the simplest case such a Hecke operator "is given by" the cohomology of the Lubin--Tate/Drinfeld space. Again, this story should extend to the $p$-adic case. The work of Hansen--Mann explains this to some extent.