TITLE: Why to believe the Fargues geometrization conjecture?
QUESTION [24 upvotes]: In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues. 
I can't even concisely state the conjecture so I will defer this to the answerers who will presumably do a much better job of it. I can, however, give a link to a paper by Fargues from which, with sufficient effort, you should be able to extract understanding of the conjecture.
Can you show us "a mathematical path" one can walk along to arrive at the conjecture, starting from the facts which you might expect an average third-year graduate student in specializing in arithmetic geometry or automorphic forms to know? A sort of a one-page (or so) explanation of why one should believe the geometrization conjecture. It would be also nice if you explain what is the relation to the geometric Langlands theory as developed by Gaitsgory and others (my guess is that they are only similar in the name but that is just a guess).
The paper of Fargues already provides a 46-page explanation but it would much easier to follow if one has a specific picture in mind to begin with. There is probably more than one answer to this question (all roads lead to Rome!) but there are not that many people in the world who understand the geometrization conjecture so if we get at least one answer that would be fortunate.

REPLY [26 votes]: Why believe in it ? Because I said so. There is absolutely no doubt it is true.
This is in some sense evident to me now, even more than in 2014 after my talk at the MSRI, in particular after the release of our joint paper with Peter.
As said by Peter. There are numerous evidences coming from everywhere. At some point this is not a coincidence anymore and there is something as I could begin to see in 2014. Here are different steps from my point of view:

Already in my PHD work, trying to prove some particular cases of this Kottwitz conjecture describing the discrete part of the cohomology of Rapoport-Zink spaces in terms of local Langlands, I could see that Kottwitz sets are very important (see for example my conjecture that says that for each element of $B(G,\mu)$ the associated Newton stratum is non-empty (now proven in a lot of cases))
When working on those twin-towers thing I tried further to geometrize Jacquet-Langlands. I mean in some sense the isomorphism (with modern notations) $[\Omega^\diamond / \underline{\mathrm{GL}_n (\mathbb{Q}_p)]}\cong [\mathbb{P}^{n-1,\diamond} / \underline{\mathrm{D}^\times}]$ is a geometric form of Jacquet-Langlands. I did not know what to do with this, tried a few things but this was hopeless (forget about this now). Btw I really understood at this point that those Hodge-Tate periods are a big thing, see further the link with modifications of vector bundles on the curve.
Then there there was this curve thing. It was clear during my work with Fontaine that modifications of vector bundles on the curve are a key thing in the theory. I perfectly remember being super excited when I understood that the Hodge filtration of a filtered $\varphi$-module allows you to modify a vector bundle on the curve (see the talk I gave at Fontaine's conference where I say that we apply a Hecke correspondence (which gave rise to protestations by Fontaine, of course)). It was more and more clear that modifications of vector bundles were a key thing. In particular I really tried to understand this in my article at Laumon's conference: certain modification of vector bundles are the same as local Shtukas !!!! (aka BKF modules). This was later confirmed by the work of Scholze and Weinstein that says that Rapoport-Zink spaces are moduli of modifications of vector bundles. But just pronouncing the word Shtuka is a great indication that we are in a good direction : some Hecke property (aka modifications of vector bundles) is linked to moduli of Shtukas ;)
There is this property that the curve is geometrically connected and thus its $\pi_1$ is $\mathrm{Gal} ( \overline{\mathbb{Q}}_p | \mathbb{Q}_p)$...this is the starting point I took at the MSRI in 2014 : let's say there is a geometric Langlands, let's writte the Hecke property...you find Kottwitz conjecture describing the discrete part of the cohomology of local Shtuka moduli spaces...
The work of Kottwitz and Kaletha was fundamental at some point to try to link this to the classical local Langlands as formulated by Kottwitz and Kaletha.
The work of Caraiani and Scholze is fundamental too to me, I really wanted to understand were does this perverse sheaf come from ? Coupled with the understanding of the cohomology of Igusa varieties (Harris-Taylor, Mantovan, Shin, Caraiani-Scholze)
One great thing was the classification of G-bundles on the curve and realizing this is Kottwitz set B(G). This is super striking, in particular when you study reduction of G-bundles.

There are plenty of other things that come to my mind. But since the release of my joint paper with Peter I am definitely convinced. In particular the categorical conjecture is more or less evident. The functor
\begin{align*}
\mathrm{Perf} ( \mathrm{LocSys}_{\hat{G}_{\overline{\mathbb{Q}}_\ell}}) & \longrightarrow D_{lis} ( \mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell ) \\
M & \longmapsto M \ast \mathcal{W}_{\psi} 
\end{align*}
defined by the spectral action
can really be understood as some kind of non-abelian Fourier transform. In some sense the Whittaker sheaf is some kind of kernel. Then the geometric conjecture states that this extends (you define the Fourier transform on some subdomain and hope this "extends by continuity") to an equivalence
$$
D^b_{coh} ( \mathrm{LocSys}_{\hat{G}_{\overline{\mathbb{Q}}_\ell}}) \xrightarrow{\sim}  D_{lis} ( \mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell )^\omega
$$
This is really a Fourier type equivalence equivalence like Fourier-Mukai or Fourier-Deligne and so on, outside of the fact that here one side is coherent and the other one is étale (like in the Fourier-Mellin transform for a torus for example). If you put for $M$ a skycraper sheaf on a discrete parameter for $\mathrm{GL}_n$ (an irreducible $Frob$-semi-simple $\ell$-adic representation of $W_E$) this non-abelian Fourier transform is computed locally on a Whittaker stack by Drinfeld ($n=2$) and Laumon more generally. The fact that this is non-zero and gives the good thing should be an analog of Frenkel-Gaitsgory-Vilonen and Gaitsgory in the function field case over a finite field.
Typicaly this geometric conjecture implies that the two possible natural definitions of the stable Bernstein center are the same, see Example X.1.6. This implies this existence of a nice kernel of functoriality too, see Example X.1.7..
Anyway, this conjecture is true, no doubt. I think that one reason why peoples have difficulties to understand why this has to be true is that they are too focused on classical geometric Langlands that is weird thing to me in some sense. But I think at the end it may be possible that the Langlands program has to be retaken from the beginning, typically the good object locally at a place $p$ are not smooth representations $\pi$ of $G(E)$ but complexes $A\in D_{lis} (\mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell)$. As you can see the kernel of functoriality is defined at the level of $D_{lis}$ naturally. It may be possible that in the next years the Langlands program has to be re-though from the beginning, maybe the good objects that can be transferred "naturally" are not automorphic representations but some object in some $D_{lis}(whatever)$.
Stay tuned for this.