Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader knows what the "1970s version of the local Langlands conjectures" are when writing this question—there are plenty of references that will get us this far (I give one below that works in the generality I'm interested in).
So let $F$ be a finite extension of $\mathbf{Q}_p$, let $G$ be a connected reductive group over $F$, let $\widehat{G}$ denote the complex dual group of $G$ (a connected complex Lie group) and let ${}^LG$ denote the $L$-group of $G$, the semi-direct product of the dual group and the Weil group of $F$ (formed using a fixed algebraic closure $\overline{F}$ of $F$).
Here is the "standard", or possibly "standard in the 1970s", way of formulating what local Langlands should say (for more details see Borel's paper "Automorphic $L$-functions", available online (thanks AMS) here at the AMS website. One defines sets $\Phi(G)$ ($\widehat{G}$-conjugacy classes of admissible Weil-Deligne representations from the Weil-Deligne group to the $L$-group [noting that "admissible" includes assertions about images only landing in so-called "relevant parabolics" in the general case and is quite a subtle notion]) and $\Pi(G)$ (isomorphism classes of smooth irreducible admissible representations of $G(F)$), and one conjectures:
LOCAL LANGLANDS CONJECTURE (naive form): There is a canonical surjection $\Pi(G)\to\Phi(G)$ with finite fibres, satisfying (insert list of properties here).
See section 10 of Borel's article for the properties required of the map.
Now in recent weeks I have had two conversations with geometric Langlands type people both of whom have mocked me when I have suggested that this is what the local Langlands conjecture should look like. They point out that studying some set of representations up to isomorphism is a very "coarse" idea nowadays, and one should reformulate things category-theoretically, considering Tannakian categories of representations, and relating them to…aah, well there's the catch. Looking back at what both of them said, they both at a crucial point slipped in the line "well, now for simplicity let's assume we're in the function field/geometric setting. Now…" and off they went with their perverse sheaves. The happy upshot of all of this is that now one has a much better formulation of local Langlands, because one can demand much more than a canonical surjection with finite fibres, one can ask whether two categories are equivalent.
But I have been hoodwinked here, because I am interested in $p$-adic fields. So yes yes yes I'm sure it's all wonderful in the function field/geometric setting, and things have been generalised beyond all recognition. My question is simply:
Q) Can we do better than the naive form of Local Langlands (i.e. is there a stronger statement about two categories being equivalent) when $F$ is a p-adic field?
The answer appears to be "yes" in other cases but I am unclear about whether the answer is yes in the $p$-adic case. Even if someone were to be able to explain some generalisation in the case where $G$ is split, I am sure I would learn a lot. To be honest, I think I'd learn a lot if someone could explain how to turn the surjection into a more bijective kind of object even in the case of $SL(2)$. Even in the unramified case! That's how far behind I am! As far as I can see, the Satake isomorphism gives only a surjection in general, because there is more than one equivalence class of hyperspecial maximal compact in general.
Best Answer
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
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":
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.
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.