Now that our paper [Geometrization of the local Langlands correspondence](https://arxiv.org/abs/2102.13459) with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally
1) an actual conjecture, in the sense that it asks for *properties* of a given construction, not for a *construction*;
2) of a form as in geometric Langlands, in particular about an *equivalence of categories*, not merely a bijection of irreducibles.

First, I should say that in the notation of the OP, we construct a canonical map $\Pi(G)\to \Phi(G)$, and prove some properties about it. However, we are not able to say anything yet about its fibres (not even finiteness).

Moreover, we give a formulation of local Langlands as an equivalence of categories, and (essentially) construct a functor in one direction that one expects to realize the equivalence. In particular, this nails down what the local Langlands correspondence should be, it "merely" remains to establish all the desired properties of it.

Let me briefly state the main result here. Let $\mathrm{Bun}_G$ be the stack of $G$-bundles on the Fargues--Fontaine curve. We define an ($\infty$-)category $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$ of $\ell$-adic sheaves on $\mathrm{Bun}_G$. The stack $\mathrm{Bun}_G$ is stratified into countably many strata enumerated by $b\in B(G)$, and on each stratum, the category $\mathcal D(\mathrm{Bun}_G^b,\overline{\mathbb Q}_\ell)$ is the derived ($\infty$-)category of smooth representations of the group $G_b(F)$. In particular, for $b=1$, one gets smooth representations of $G(F)$.

Moreover, there is an Artin stack $Z^1(W_E,\hat{G})/\hat{G}$ of $L$-parameters over $\overline{\mathbb Q}_\ell$.

Our main theorem is the construction of the "spectral action":

> Theorem. There is a canonical action of the $\infty$-category of perfect complexes on $Z^1(W_E,\hat{G})/\hat{G}$ on $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$.

The main conjecture is basically that this makes $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^\omega$ a "free module of rank $1$ over $\mathrm{Perf}(Z^1(W_E,\hat{G})/\hat{G})$", at least if $G$ is quasisplit (or more generally, has connected center).

More precisely, assume that $G$ is quasisplit and fix a Borel $B\subset G$ and a generic character $\psi$ of $U(F)$, where $U\subset B$ is the unipotent radical, giving the Whittaker representation $c\text-\mathrm{Ind}_{U(F)}^{G(F)}\psi$, thus a sheaf on $[\ast/G(F)]$, which is the open substack of $\mathrm{Bun}_G$ of geometrically fibrewise trivial $G$-bundles; extending by $0$ thus gives a sheaf $\mathcal W_\psi\in \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$, called the Whittaker sheaf.

> Conjecture. The functor
$$ \mathrm{Perf}(Z^1(W_E,\hat{G})/\hat{G})\to \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$$
given by acting on $\mathcal W_\psi$ is fully faithful, and extends to an equivalence
$$\mathcal D^{b,\mathrm{qc}}_{\mathrm{coh}}(Z^1(W_E,\hat{G})/\hat{G})\cong \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^{\omega}.$$

Here the superscript $\mathrm{qc}$ means quasicompact support, and $\omega$ means compact objects. As $Z^1(W_E,\hat{G})$ is not smooth (merely a local complete intersection), there is a difference between perfect complexes and $\mathcal D^b_{\mathrm{coh}}$, and there is still a minor ambiguity about how to extend from perfect complexes to all complexes of coherent sheaves. Generically over the stack of $L$-parameters, there is however no difference.

It takes a little bit of unraveling to see how this implies more classical forms of the correspondence, like the expected internal parametrization of $L$-packets; in the case of elliptic $L$-parameters, everything is very clean, see Section X.2 of our paper.

(There are related conjectures and results by [Ben-Zvi--Chen--Helm--Nadler](https://arxiv.org/abs/2010.02321), [Hellmann](https://arxiv.org/abs/2006.03013) and [Zhu](https://arxiv.org/abs/2008.02998); see also the work of [Genestier--Lafforgue](https://arxiv.org/abs/1709.00978) in the function field case. And this work is heavily inspired by previous work in geometric Langlands, notably the conjectures of [Arinkin--Gaitsgory](https://arxiv.org/abs/1201.6343), and the work of [Nadler--Yun](https://arxiv.org/abs/1611.04078) and [Gaitsgory--Kazhdan--Rozenblyum--Varshavsky](https://arxiv.org/abs/1908.05420) on spectral actions.)

PS: It may be worth pointing out that this conjecture is, at least a priori, of a quite different nature than Vogan's conjecture, mentioned in the other answers, which is based on perverse sheaves on the stack of $L$-parameters; here, we use coherent sheaves.