Possibly an easy application of Tate-Nakayama duality and local CFT Let $F$ be a non-archimedian local field and $F^{ur}$ the maximal unramified extension. Let $T$ be a torus defined over $F$ such that $T$ splits over a tamely ramified extension and is anisotropic over $F^{ur}$, i.e. $T\times_{\text{Spec} F}\text{Spec} F^{ur}$ contains no $\mathbb{G}_m$. Let $T(F)$ be the (topological) group of rational points and $T(F)_+$ be the pro-$p$ part of $T(F)$: $T(F)_+=\{g\in T(F)\;|\;\lim_{n\rightarrow\infty}g^{p^n}=id\}$
Let $\hat{T}:=X^*(T)\otimes(\mathbb{C}/\mathbb{Z})$ be the Langlands dual of $T$. I am wondering if there is a canonical isomorphism between the two abstract abelian groups 
$$(\hat{T})^{\Gamma}\cong (T(F)/T(F)_+)^{\vee}.$$
(Edited: this is likely wrong or at least not the canonical one, sorry. Please see my post below for correction)
Edited: I am thinking of this as an example of something that I want to think of as "unramified local Langlands for tamely ramified tori" (while I don‘t know what it should mean). As the title suggests I suspect it (if valid) to be some simple application of Tate-Nakayama, and/or local Langlands for tori, but have no idea what to do with the RHS. Thank you!
 A: Since this is too long for comment I'll put it here. I just realize the formulation of mine is likely incorrect. In usual language the RHS of the equality $(T(F)/T(F)_+)^{\vee}$ is the set of "tamely-ramified" or "depth-zero" representation of $T(F)$. On the other hand, the set of local Langlands parameters that is trivial on the wild inertia $P_F$ is $H^1(W_F/P_F,\hat{T})$, where $W_F$ is the Weil group of $F$ and $I_F$ is the inertia of $F$. It makes sense as we assume $T$ to be tamely ramified. We have exact sequence 
$$0\rightarrow H^1(W_F/I_F,\hat{T}^{I_F})\rightarrow H^1(W_F/P_F,\hat{T})\rightarrow H^1(I_F/P_F,\hat{T}).$$
The assumption that $T$ is anisotropic over $F^{ur}$ says the last cohomology vanishes (so only the "Frebenius" part contributes). Thus $H^1(W_F/I_F,\hat{T}^{I_F})$ is the group of tamely-ramified Langlands parameters, and we should have
$$H^1(W_F/I_F,\hat{T}^{I_F})=(T(F)/T(F)_+)^{\vee}$$
while my original guess was $\hat{T}^{\Gamma}=H^0(W_F/I_F,\hat{T}^{I_F})$, which is a group of the same order but should not be the same.
Since the local Langlands for depth-zero / tamely ramified case has been established in various content I firmly believe this should be there somewhere. But on the other hand this should really be the basic case and I'll be very appreciated if somebody shares a direct proof.
