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.