Now that our paper Geometrization of the local Langlands correspondence 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
- an actual conjecture, in the sense that it asks for properties of a given construction, not for a construction;
- 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 $Z^1(W_F,\hat{G})/\hat{G}$ of $L$-parameters over $\overline{\mathbb Q}_\ell$.
Our main result is the construction of the "spectral action":
There is a canonical action of the $\infty$-category of perfect complexes on $Z^1(W_F,\hat{G})/\hat{G}$ 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 $\mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})$", 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
$$ \mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})\to \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$$
given by acting on $\mathcal W_\psi$ is fully faithful, and extends to an equivalence
$$\mathcal D^{b,\mathrm{qc}}_{\mathrm{coh}}(Z^1(W_F,\hat{G})/\hat{G})\cong \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^{\omega}.$$
Here the superscript $\mathrm{qc}$ means quasicompact support, and $\omega$ means compact objects. As $Z^1(W_F,\hat{G})$ 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, Hellmann and Zhu; see also the work of Genestier--Lafforgue in the function field case. And this work is heavily inspired by previous work in geometric Langlands, notably the conjectures of Arinkin--Gaitsgory, and the work of Nadler--Yun and Gaitsgory--Kazhdan--Rozenblyum--Varshavsky 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.
Here is an extremely naive answer to a case of my own question, which is surely obvious to experts. For me, even coming up with this silly version required quite a bit of conversation with Sugwoo Shin (who is of course blameless of any errors).
We give a description of the odd $GL_2$-Galoisian sets of level $N$ in terms of 'higher congruence conditions.' The point is to consider the $\mathbb{Q}$-Hecke algebra $H(N)$ determined by the Hecke operators acting on modular forms of weight 1, level $N$.
The maximal ideals in $H(N)$ are in correspondence with Galois conjugacy classes of normalized new weight one eigenforms of level $N$. There is also a map $$p\mapsto T_p$$
from primes not dividing $N$ to $H(N)$.
Any given maximal ideal $m$ determines a Dirichlet character $\epsilon_m$, and one considers the set of primes $S(m)$ defined by the 'congruence conditions'
$$(p,N)=1, \ \epsilon_m(p)=1, \ T_p\equiv 2 \ \ \mod \ m$$
These $S_m$ are exactly the odd $GL_2$ Galoisian sets of level $N$.
With a bit more care, one should be able to give a similar description that doesn't refer to the level beforehand.
Of course this is no different from what I wrote before, but focussing on the Hecke algebra seems to allow a formulation that's rather analogous to the classical one. That is, one can forget about modular forms for a moment and examine sets of primes determined by congruences in the Hecke algebra.
Best Answer
I am confused by your negative reaction to a putative assertion that every Galois repn's L-function is automorphic (perhaps meaning that it is the standard L-function attached to a cuspidal automorphic form on some $GL_n$ over $\mathbb Q$, and, thus, has the expected analytic continuation and functional equation). To me, various forms of the assertion that every motivic L-function is automorphic (with what is implied...) is one of the best capsulizations of "L's conjectures".
Ok, yes, this does mostly disregard the obvious intuitive senses of "functoriality", refering to putative maps/correspondences of afms on one group to another. And, yes, we know (potential modularity etc: Harris-Taylor's Sato-Tate, et alia) that just a lil' bit of "modularity" goes a long way...
I think that combining the "raw" conjectural ideas with the other dose of conjecture, namely, about existence of a group whose tensor/Tannakian/whatever category is ... automorphic forms...?... may add enough ambiguity to leave it all tooo ambiguous. Or, perhaps, those things will provoke someone's imagination?
But, seriously, two operational components come to mind: motivic L-functions are automorphic, and, "functoriality" is valid for automorphic L-functions.