Treumann–Venkatesh's linkage conjecture for functorial transfer

Let $F_v$ be a local function field with ring of integers $\mathcal{O}_v$ and characteristic $\neq p$. Let $G_v$ be a reductive group over $F_v$ with a $\sigma$-action, let $H_v=G_v^{\sigma}$, and let $k$ be the coefficient field. Let $\Pi$ be a smooth irreducible representation of $G_v$ that is $\sigma$-fixed, so that its $G_v$-action extends to $G_v\rtimes\langle\sigma\rangle$ and its Tate cohomology groups $T^0(\Pi)$ and $T^1(\Pi)$ are representations of $H_v$. An irreducible admissible representation $\pi$ of $H_v$ is linked with an irreducible admissible representation $\Pi$ of $G_v(F_v)$ if $\pi^{(p)}$ appears in $T^0(\Pi)$ or $T^1(\Pi)$, where

$$\pi^{(p)}:=\pi\otimes_{k,\operatorname{Frob}}k.$$

Treumann–Venkatesh's linkage conjecture. Linkage is compatible with functorial transfer: if $\pi$ is linked to $\Pi$, then $\pi$ transfers to $\Pi$ under the Local Langlands correspondence.

The conjecture proposes compatibility between Tate-cohomological linkage and local Langlands functoriality. Treumann–Venkatesh also conjecture that the Tate cohomology groups are admissible representations of $H_v$, but that separate assertion is not part of this linkage conjecture and is not proved or used here.