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.