Local Langlands correspondences in $\ell$-adic coefficients

Let $\ell$ be a prime number different from the residue characteristic of a non-archimedean local field $F$. We give formulations of $\ell$-adic local Langlands correspondences for connected reductive algebraic groups over $F$, which we conjecture to be independent of a choice of an isomorphism between the $\ell$-adic coefficient field and the complex number field.