Background:
(Mostly my paraphrased interpretation of the introduction of Strauch's Deformation spaces of one-dimensional formal modules and their cohomology, with additional details from Carayol's Non-Abelian Lubin-Tate theory. Then some comments from parts of works of Kottwitz, Harris-Taylor, Scholze, and Chojecki-Knight cited later.)
Let $F := \mathbb{F}_q((T))$ or $\mathbb{Q}_p$. Let $O_F$ be its ring of integers, let $B$ be the quarternion division algebra over $F$, and $W_F$ is the Weil group of $F$.
Carayol studied the bad reduction of deformation spaces of one-dimensional formal modules and their cohomology in On the bad reduction of Shimura curves. In the paper On adic-representations associated to Hilbert modular forms he constructed a representation of $GL_2(F) \times B^\times \times W_F$ in terms of vanishing cycles at singular points in the special fibre at primes of bad reduction of these curves. This representation is constructed purely locally and its existence rests solely on the representability of the deformation spaces with level structures, and can hence be carried out in the equal characteristic case too.
The importance of this representation is that it realizes simultaneously the Jacquet-Langlands and the Langlands correspondence. It is of the form
$$Rep := \bigoplus \pi \otimes \rho \otimes \sigma$$
where $\pi$, $\rho$, and $\sigma$ are $\overline{\mathbb{Q}}_\ell$-representations of $GL_2(F)$, $B^\times$, and $W_F$ respectively; and a tensor product $\pi \otimes \rho \otimes \sigma$ occurs iff $\pi$ and $\rho$ correspond under Jacquet-Langlands, and $\pi$ and $\sigma$ are related by local Langlands.
Fix the height $h$. Since we can equip Lubin-Tate space $LT$ with $m$-level structure, a summary of Carayol's generalization via vanishing cycles is as follows.
Let $Y_{m,h}$ be the classifying space of level $m$ structures on a universal group law over the generic fiber of Lubin-Tate space. We can naturally endow $Y_{m, h} \otimes_{\hat{F}^{nr}} \hat{\overline{F}}$ with a rigid structure. We define its associated representation as:
$$U_h^v := \text{colim}_m \text{ } H^{h-1}_{et}(Y_{m, h} \otimes_{\hat{F}^{nr}} \hat{\overline{F}}, \overline{\mathbb{Q}}_{\ell})$$
He gets the action of $GL_m(F) \times B^\times \times W_F$ on $Y_{m, h}$ by first taking the natural action of $GL_m(\mathcal{O}_F) \times \mathcal{O}_{B^\times} \times \text{Gal}(\overline{F}/F^{nr})$, and extending it a little to a larger subgroup, then inducing up to the full group.
Then, Carayol sketches the proof for h=2 that $$U_h^v \simeq Rep$$
as representations of $GL_h(F) \times B^\times \times W_F$, for all $m \in \mathbb{N}$, but only for $\ell \neq p$. He then conjectures it to be true for $h \geq 3$.
Carayol also does the same procedure for $h=2$ in the "rigid setting", replacing $Y_{m,h}$ with Drinfel'd's coverings of the p-adic upper half plane $\Sigma_{m, h}$. The only difference is that the variance of $\rho$ changes to $\rho^\vee$.
We go through the definition of $\Sigma_{m, h}$ here. Let $G$ be the universal formal $O_B$-module over the deformation space of a fixed formal $O_B$-module. Let $\mathcal{G}[m] := G[\omega^m] \otimes_{\hat{O}^{nr}} \hat{F}^{nr}$, the rigidified $\omega^m$ torsion points of $G$. Then we take the points which are of order exactly $\omega^m$.
$$\Sigma_{m, h} := \mathcal{G}[m] - \mathcal{G}[m-1]$$
Kottwitz in Points of some Shimura varieties over finite fields introduces some ways of analyzing cohomology of PEL Shimura varieties by classifying and then counting isogeny classes.
Reading through Carayol's proof, he seems to use more than just the representability of the deformation spaces with level structures. The first part, I could believe comes from representability: He constructs $K$ such that the analytified Shimura variety $S_K$ base changed to $\mathbb{Q}_p$ is isomorphic to $(\Sigma_{m, h} \times X_K)/GL_2(\mathbb{Q}_p)$, for an appropriate coset $X_K$. (Is this why $\Sigma_{m,h}$ and $Y_{m, h}$ are sometimes called local Shimura varieties?) He then goes through a global-to-local argument. He deduces cohomology of $S_K \otimes \mathbb{Q}_p$ by calculating the LHS and RHS of: $$0 \to H^1(\Gamma,H^0(X; \overline{\mathbb{Q}}_\ell)) \to H^1(X/\Gamma, \overline{\mathbb{Q}}_\ell) \to H^1(X/\Gamma, \overline{\mathbb{Q}}_\ell) \to 0$$ for $X = S_K \otimes \mathbb{Q}_p$, $\Gamma = GL_2(\mathbb{Q}_p)$ which comes from the Cartan-Leray spectral sequence. Finally, he uses some results about realizing the correspondence in $H^0$ to shimmy it up to $H^1$. In this last step, he uses more than just the representability of the deformation spaces with level structures, does he not?
Harris and Taylor in On the geometry and cohomology of some simple Shimura varieties (page 8) seem to then realize local Langlands for $GL_h(F)$ for all $h$ at primes of bad reduction $\ell \neq p$ (see end of page 4). They use $U(1, h-1)$ Shimura variety in place of $S_K$, and a global-to-local argument. They of course do much much more, including using virtual representations. They seem to resolve here Carayol's conjectures for $h\geq 3$, $\ell \neq p$, if I understand correctly.
Scholze then simplified part of their work in a series of three papers GL2 GLn and RZ.
In the third paper RZ, he uses the representability results of the "PEL/isocrystal-built" moduli space of p-divisible groups of Rapoport-Zink in Period Domains for p-divisible groups (pg. 79). He seems to then compute directly that $Rep$ is isomorphic to the cohomology of this moduli space at primes of bad reduction (with again the assumption $\ell \neq p$, see theorem 2.1), with the help of the Hodge-Tate period map, without using a global-to-local argument.
The case of $\ell = p$, there is a candidate Knight's thesis for h=2, and more generally in Chojecki-Knight's p-adic Jacquet-Langlands correspondence and patching from 2017.In the latter, they use etale cohomology (defined on page 5). Let $E$ be a finite extension of $\mathbb{Q}_p$, where $\omega$ is the uniformizer of $E$. They look at: $$\lim_s \text{colim}_{K_p} \text{ }H^*_{et}(S_{K_pK^p}; O_E/\omega^s)$$ In our previous overview of $\ell \neq p$, $K = K_pK^p$ as well. This almost looks like cristalline cohomology, except we are not considering $S_{K_pK^p}$ as a cristalline topos over $\text{Spec } W_n(k)$.
Preliminary question: Is my understanding of the background correct? I had quite a bit of difficulty properly digesting, and am worried I got something wrong.
Morally, I have the feeling that taking equicharacteristic etale cohomology is very destructive, and it is better to take, for example, cristalline cohomology. Further, it seems that Lubin-Tate space of height $h$ with infinite level structure is unavoidably entrenched in p-adic Hodge theory, since it (1) is naturally a pro-etale Galois cover of $\mathbb{P}^{h-1}(F)$, and (2) can be fully-faithfully embedded into the category of diamonds.
My Question: What is known about Carayol's conjecture when $\ell = p$? $$\text{colim}_m \text{ } H^{h-1}_{et}(Y_{h, m} \otimes_{\hat{F}^{nr}} \hat{\overline{F}}, \overline{\mathbb{Q}}_{p}) \stackrel{?}{\simeq} Rep$$
Does it hold when we take cristilline cohomology instead of rigid $\ell$-etale cohomology?
I repeat for convenience the definiton of $Rep$.
$$Rep := \bigoplus \pi \otimes \rho \otimes \sigma$$
where $\pi$, $\rho$, and $\sigma$ are $\overline{\mathbb{Q}}_\ell$-representations of $GL_2(F)$, $B^\times$, and $W_F$ respectively; and a tensor product $\pi \otimes \rho \otimes \sigma$ occurs iff $\pi$ and $\rho$ correspond under Jacquet-Langlands, and $\pi$ and $\sigma$ are related by local Langlands.
