TITLE: Langlands correspondence for higher local fields?
QUESTION [21 upvotes]: Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible admissible representations of $GL_n(F)$ and the set of equivalence classes of continuous Frobenius semisimple complex $n$-dimensional Weil-Deligne representations of the Weil group of $F$ that preserves $L$-functions and $\epsilon$-factors. This statement has been proven for all one-dimensional local fields. 
My question is: is it possible to formulate any meaningful analogue of Langlands conjectures for higher local fields (e.g. formal Laurent series over $\mathbb{Q}_p$)? If this is possible, conjectures for $GL_1$ should probably be equivalent to higher local class field theory (it says that for an $n$-dimensional local field $F$, there is a functorial map
$$
K_n(F)\rightarrow \mathrm{Gal}(F^{ab}/F)
$$
from $n$-th Milnor K-group to the Galois group of maximal abelian extension, which induces an isomorphism $K_n(F)/N_{L/F}(K_n(L))\rightarrow \mathrm{Gal}(L/F)$ for a finite abelian extension $L/F$).
Frankly, I do not even know what should be the right definition of Weil group of a higher local field (nor did my literature search give any results) but maybe other people have figured it out.

REPLY [4 votes]: Similar and related questions are: 
On Geometric Langlands Correspondence
Langlands conjectures in higher dimensions
Kapranov's analogies
probably more...

I've already wrote on subj here: https://mathoverflow.net/q/131884
Let me just add a few words.
1) Abelian case of higher dimensional Langlands (=class field theory) developped by A.N. Parshin (1975) and K.Kato (1977) and later on by Fesenko and others (survey 2000). 
2) Around 1992-5 Mikhail Kapranov wrote quite a speculative paper "Analogies between the Langlands correspondence and topological quantum field theory"
(See: Kapranov's analogies )
One of his ideas is the following:
Instead correspondence between Rep(Gal) <-> Rep( G(Adelic) ),
one should consider "higher represenations" (representations not into category of vector spaces, but to k-categories). 
So Kapranov's idea:  n-th dimensional k-representations of dimension r of Galois group should correspond to (n-k)-representations of GL_r(n-Local Field)
That will include Parshin-Kato abelian case as a subcase as Kapranov explained.
However actual work with higher representations is somewhat elusive, 
so difficult to transform insights into precise theorems/conjectures.